Qutbism and Terrorism
Qutbism (also Kotebism, Qutbiyya, or Qutbiyyah) is a strain of Islamist ideology and activism, based on the thought and writings of Sayyid Qutb, an Islamist and former leading member of the Muslim Brotherhood who was executed in 1966. It has been described as advancing the concept of "offensive jihad," - waging jihad in conquest - or "armed jihad in the advance of Islam" 
Qutbism has gained notoriety from what many believe is his strong influence on jihadi extremists such as Osama bin Laden. According to observers, jihadi extremists “cite Sayyid Qutb repeatedly and consider themselves his intellectual descendants.”
Qutbee or Qutbi (also Qutbists) are followers of these ideals. These terms originated from, and are mainly used by opponents of the Muslim Brotherhood in general and Qutb in particular, and by Muslims who seek to distance themselves from the activities of militant groups based on or influenced by Qutbism.
The main tenet of Qutbist ideology is that the Muslim community (or the Muslim community outside of a vanguard fighting to reestablish it) "has been extinct for a few centuries"  having reverted to Godless ignorance (Jahiliyya), and must be re-conquered for Islam.
Qutb outlined his ideas in his book Ma'alim fi-l-Tariq (aka Milestones). Other important principles of Qutbism include
adherence to Sharia as sacred law accessible to humans, without which Islam cannot exist
adherence to Sharia as a complete way of life that will bring not only justice, but complete freedom from servitude, peace, personal serenity, scientific discovery and other benefits;
avoidance of Western and non-Islamic "evil and corruption," including socialism and nationalism;
vigilance against Western and Jewish conspiracies against Islam
a two-pronged attack of 1) preaching to convert and 2) jihad to forcibly eliminate the "structures" of Jahiliyya.
the importance of offensive Jihad to eliminate Jahiliyya not only from the Islamic homeland but from the face of the earth.
a fusion of puritanical and intolerant Islamic orientations," that includes not only Qutb's ideas but those of Abul Ala Maududi, Hassan al Banna, and even Shia elements,
"to justify armed jihad in the advance of Islam, and other violent methods utilized by twentieth century militants. ... Qutbism advocates violence and justifies terrorism against non-Muslims and apostates in an effort to bring about the reign of God. Others, i.e., Ayman Al-Zawahiri, Abdullah Azzam, and Osama bin Laden built terrorist organizations based on the principles of Qutbism and turned the ideology of Islamic-Fascism into a global action plan."
 History of the word "Qutbee"
Following Qutb's death Qutbist ideas spread throughout Egypt and other parts of the Arab and Muslim world, prompting a backlash by more traditionalist and conservative Muslims, such as the book Du'ah, la Qudah (Preachers not Judges), (1969), written by the MB Supreme Guide Hasan al-Hudaybi.
The word Qutbee is said to have first been used by Saudi Arabian Salafist Muslims to refer not only to explicit devotees of Qutb's ideas, but to Muslim Brotherhood members and their sympathizers in general(despite the fact that there is a range of opinion among Muslim Brethren on Qutb's ideas.) The word Qutbee is used in a similar way as the term Wahhabi in that it is used not by the individuals it describes themselves, but by their opponents.
The most controversial aspect of Qutbism is Qutb's idea that Islam is "extinct," and therefore those who say they are Muslims--with the exception of Qutb's Islamic vanguard--are not. Intended to shock Muslims into religious re-armament it also had the effect, if taken literally, of making non-Qutbists who claimed they were Muslims in serious violation of the traditional Sharia law (law that Qutb very much supported) potentially finding them guilty of Apostasy in Islam and their blood fit to be shed.
Because of these serious consequences, Muslims have traditionally been reluctant to practice takfir, that is, to pronounce professed Muslims as unbelievers (even Muslims in violation of Islamic law). This prospect of fitna, or internal strife, between Qutbists and "takfir-ed" mainstream Muslims, was put to Qutb by prosecutors in the trial that led to his execution,  and is still made by his Muslim detractors. 
Qutb died before he could clear up the issue of whether jahiliyya referred to the whole "Muslim world," to only Muslim governments, or only in an allegorical sense. But that his critics fear of fitna was legitimate would seem to have been borne out by a serious campaign of Islamist terror - or what Qutb might have called "physical power and jihad" against "the organizations and authorities" of "jahili" Egypt - in the 1980s and 1990s.
Victims included Egyptian President Anwar Sadat, head of the counter-terrorism police Major General Raouf Khayrat, parliamentary speaker Rifaat el-Mahgoub, dozens of European tourists and Egyptian bystanders, and over one hundred Egyptian police. Other factors, (such as economic dislocation/stagnation and rage over President Sadat's policy of reconciliation with Israel) played a part in instigating the violence, but Qutb's takfir against jahili society, and his passionate belief that Jahiliyya government was irredeemably evil and must be destroyed played a key role.
 Traditionalist criticism
While Ma'alim fi-l-Tariq (Milestones) was Qutb's manifesto, other elements of Qutbism are found in his works Al-'adala al-Ijtima'iyya fi-l-Islam (Social Justice in Islam), and his Quranic commentary Fi Zilal al-Qur'an (In the shade of the Qur'an). Ideas in (or alleged to be in) those works also have come under attack from traditionalist/conservative/Wahhabi Muslims. They include
Qutb's assertion that slavery is now illegal under Islam, as its lawfulness was only temporary, existing only "until the world devised a new code of practice, other than enslavement." Traditionalist critics maintain
Islaam has affirmed slavery ... And it will continue so long as Jihaad in the path of Allaah exists."(Shaikh Salih al-Fawzaan) 
Proposals to redistribute income and property to the needy. Opponents claim they are "socialist" and innovations of Islam.  (Though Qutb was in favor of "social justice", he strongly disapproved of socialism - even of "Islamic socialism" - seeing it as compromise with jahiliyya.
Describing Moses as having an "excitable nature" - this allegedly being "mockery," and "mockery of the Prophets is apostasy in its own,'" according to Shaikh ‘Abdul-Azeez Ibn Baz.
Dismissing fiqh or the schools of Islamic law known as madhhab as separate from "Islamic principles and Islamic understanding."
Desiring to unite the four schools of Islamic law into one school - allegedly an innovation.
Favoring the overthrow of tyrants, when Islam teaches that "when you cannot correct a wrong thing be patient! Allah ... will correct it."
Accusations against Qutbism include some that may be very questionable, such as one alleging that Qutb believed "Christians should be left as Christians--Jews as Jews," since he believed in hurriyatul-i'tiqaad (freedom of belief) 
To some extent these attacks may represent Qutbism's success or its logical conclusion as much as its failure to persuade some critics. Qutb sought Islamically-justified alternatives to European ideas like Marxism and socialism and proposed Islamic means to achieve the ends of social justice and equality, redistribution of private property, political revolution.
Many of his critics want to replace not just Western means but ends as well. "Neofundamentalist refuse to express their views in modern terms borrowed from the West.
They consider that indulging in politics, even for a good cause, will by definition lead to bid'a and shirk (the giving of priority to worldly considerations over religious values.)" 
There are, however, some commentators who display an ambivalence towards him, noting that "his books are found everywhere and mentioned on most neo-fundamentalist websites, and arguing his "mystical approach" and "pessimistic views on the modern world" have resonated with some Muslims.
 Science and learning
On the importance of science and learning, the key to the power of his bete noire, western civilization, Qutb was ambivalent. He wrote that
Muslims have drifted away from their religion and their way of life, and have forgotten that Islam appointed them as representatives of God and made them responsible for learning all the sciences and developing various capabilities to fulfill this high position which God has granted them.
... and encouraged Muslims to seek knowledge.
A Muslim can go to a Muslim or to a non-Muslim to learn abstract sciences such as chemistry, physics, biology, astronomy, medicine, industry, agriculture, administration [disambiguation needed] (limited to its technical aspects), technology, military arts and similar sciences and arts; although the fundamental principle is that when the Muslim community comes into existence it should provide experts in all these fields in abundance, as all these sciences and arts are a sufficient obligation (Fard al-Kifayah) on Muslims (that is to say, there ought to be a sufficient number of people who specialize in these various sciences and arts to satisfy the needs of the community). (Qutb, Milestones p.109)
On the other hand, Qutb believed some learning was forbidden to Muslims and should not be studied, including:
principles of economics and political affairs and interpretation of historical processes ... origin of the universe, the origin of the life of man ... philosophy, comparative religion ... sociology (excluding statistics and observations) ... Darwinist biology ([which] goes beyond the scope of its observations, without any rhyme or reason and only for the sake of expressing an opinion ...). (Qutb, Milestones p.108-110)
and that the era of scientific discovery (that non-Muslim Westerners were so famous for) was now over:
The period of resurgence of science has also come to an end. This period, which began with the Renaissance in the sixteenth century after Christ and reached its zenith in the eighteenth and nineteenth centuries, does not possess a reviving spirit. [Qutb, Milestones p.8]
However important scientific discovery was or is, an important tool to achieve it (and to do everything else) is to follow Sharia law under which
blessings fall on all mankind, [and] leads in an easy manner to the knowledge of the secrets of nature, its hidden forces and the treasures concealed in the expanses of the universe. [Qutb, Milestones p.90]
 Qutbism and non-Muslims
Other elements of Qutbism deal with non-Muslims, particularly Westerners, and have drawn attention and controversy from their subjects, particularly following 9/11. Though their terminology, issues and arguments are different from those of the Islamic traditionalists, Westerners also have criticism to make.
 Islamic law and freedom
Qutbism postulates that sharia-based society will have an almost supernatural perfection, providing justice, prosperity, peace and harmony both individually and societally. 
Its wonders are such that the use of offensive jihad to spread of sharia-Islam throughout the non-Muslim world will not be aggression but "a movement ... to introduce true freedom to mankind." It frees humanity from servitude to man because its divine nature requires no human authorities to judge or enforce its law. 
 Vigilance against conspiracies
Qutbism emphasizes the (alleged) evil designs of Westerners and Jews against Islam, and the importance of Muslims not trusting or imitating them.
 The West
In Qutb's view, for example, Western Imperialism is not, as (leftist) Westerners would have Muslims believe, only an economic exploitation of weak peoples by the strong and greedy. Nor were the medieval Crusades, as some historians claim, merely an attempt by Christians to reconquer the formerly Christian-ruled, Christian holy land.
Both were different expressions of the West's "pronounced ... enmity" towards Islam, including plans to "demolish the structure of Muslim society."  Imperialism is "a mask for the crusading spirit." 
Examples of Western malevolence Qutb personally experienced and related to his readers include an attempt by a "drunken, semi-naked ... American agent" to seduce him on his voyage to America, and the (alleged) celebration of American hospital employees upon hearing of the assassination of Egyptian Ikhwan Supreme Guide Hasan al-Banna.
Qutb's Western critics have questioned whether Qutb was likely to arouse interest of American intelligence agents (as he was not a member of the Egyptian government or any political organization at that time), or whether many Americans, let alone hospital employees, knew who Hasan al-Banna or the Muslim Brotherhood were in 1948.
The other anti-Islamic conspirator group, according to Qutb, is "World Jewry," which he believes is engaged in tricks to eliminate "faith and religion", and trying to divert "the wealth of mankind" into "Jewish financial institutions" by charging interest on loans.
Jewish designs are so pernicious, according to Qutb's logic, that "anyone who leads this [Islamic] community away from its religion and its Quran can only be [a] Jewish agent", causing one critic to claim that the statement apparently means that "any source of division, anyone who undermines the relationship between Muslims and their faith is by definition a Jew".
 Western corruption
Qutbism emphasizes the importance of Islamic superiority over the West. One example of "the filth" and "rubbish heap of the West." (Qutb, Milestones, p.139) was the "animal-like" "mixing of the sexes." Qutb alleges that while he was in America a young woman told him
The issue of sexual relations is purely a biological matter. You ... complicate this matter by imposing the ethical element on it. The horse and mare, the bull and the cow ... do not think about this ethical matter ... and, therefore live a comfortable, simple, and easy life. 
Critics complain that this opinion was wildly unrepresentative and the incident highly improbable. Even at the height of the sexual revolution in America 30 years later, most Americans would disagree with his statement, but at the time of his visit to America, sex out of wedlock, let alone "animal-like" promiscuity, was rare, with the overwhelming number of Americans married as virgins or only had premarital sex with their future spouse.
 Muslim Brotherhood
Controversy over Qutbism is in part an expression of the disagreement of two of the main tendencies of the Islamic revival: the more traditional Salafi Muslims, and the more radically active Muslim groups associated with the Muslim Brotherhood, the group Qutb was a member of for about the last decade and a half of his life.
Although Sayyid Qutb was never head (or "Supreme Guide") of the Muslim Brotherhood, he was the Brotherhood's "leading intellectual,"  editor of its weekly periodical, and a member of the highest branch in the Brotherhood, the Working Committee and of the Guidance Council.
After the publication of Ma'alim fi-l-Tariq, (Milestones), opinion in the Brotherhood split over his ideas, though many in Egypt (including radicals outside the Brotherhood) and most Brethren in other countries are said to have shared his analysis "to one degree or another."  In recent years his ideas have been embraced by radical Islamists groups while the Muslim Brotherhood has tended to serve as the official voice of Islamist moderation.
Islamic Extremism Jamaat-i-Islami
Maulana Sayyid Abul Ala Maududi, born in India in 1903, became perhaps one of the most influential writers and thinkers in twentieth century Islamic politics. Even those who disagree with him are unable to escape the basic concepts and terminology which he created - for example, "Islamic movement," "Islamic politics," and "the Islamic system of life."
His basic goal was to make Islam the supreme organizing principle in the social andpolitical life of the Muslim ummah. The concept upon which he based this was iqamat-i-deen, which literally means "the establishment of religion." According to this idea, all institutions of civil society and the state must be totally subordinated to the authority of divine law as revealed in the Qur'an and practiced by Muhammad.
Thus, it was insufficient for members of society to practice Islam in their personal lifes, no matter how devoutly and consistently. Instead, faith must manifest itself in the social, economic and political spheres of society. In practice, this meant establishing the shari'a as the basic law of the land which all citizens must follow, as well as the elimination of man-made civil law as an abomination to God's sovereignty.
It is neither for us to decide the aim and purpose of our existence nor to prescribe the limits of our worldly authority, nor is anyone else entitled to make these decisions for us. ...Nothing can claim sovereignty, be it a human being, a family, a class, or a group of people, or even the human race in the world as a whole. God alone is the Sovereign, and His commandments the Law of Islam.
Thus, no ruler who refused to fully institute Islamic law really deserved obedience. Secular rulers and their secular laws do not have authority over the believer - only God and God's laws do. Revolution is not simply a right, but is in fact a duty of every Muslim.
For this reason, Muslims must engage in jihad in order to re-create their society and ultimately the world. He even declared that jihad was a central tenet of Islam, on par with the traditional Five Pillars of Faith. Maududi became perhaps the first systematic Islamic thinker to to put jihad in such a central religious position.
In 1941, Maududi founded the Jamaat-i-Islami ("Islamic Party") to give his program an institutional form and movement. He hoped to reconstitute Indian society based upon Islamic law, although most members ended up moving to Pakistan when that new state was created in order to ensure that Pakistan was made and would stay a Muslim society.
For Maududi, Muslims lived at a time of crisis: Islam was in serious danger of being destroyed by the power of the West. Because of this, Muslims could not retreat from political life and leave matters in the hands of secular politicians - they must instead take an active part in politics to ensure the survival of Islam, and to keep secular politicians from coming to power in the first place.
Doing this was also designed to help ensure their personal survival, too. Secular rulers were corrupt and immoral - this is a consequence of preferring the laws of Man over the laws of God. Instituting Islamic law, however, would ensure that corrupt rulers would never come to power, and that therefore society would be just and equitable for all.
Fast-forward 60 years. On October 19th, 2001, the leader of the Jamaat, ameer Qazi Hussain Ahmad, declared that the Pakistani government should be overthrown if it does not cease its support of Western governments against Afghanistan. According to the Jamaat, the conflict between the West and Afghanistan is not over terrorism, but over religion.
Undoubtedly the real issue - which unfortunately a few understand - is the status of testimony of the Prophethood of Muhammad. ...Intent to enslave Muslim rulers had firmly been established now when we see a vast network of Western army installations in the Middle East, which negotiate nothing besides weakening the Muslim countries in all
respects. ...The Westís fathering the human rights, the emancipation of women and the like - slogans raised in the name of civility - seem malicious and reflect religious bigotry. ...Under the lucrative so-called slogans of human rights, status of women and democratic governance, etc., this group is injecting sugarcoated poison in peoples' minds and thus trying to re-shape the life-style of the Ummah. In course of time, they believe, unshakable Muslim faith on the absolute truth and the prophethood shall gradually and automatically terminate this way.
All of the above ideas have their origins in the writings of Maududi. In 1951, Maududi's works began to be published in Egypt just at the time when agitation towards increased Islamization was beginning. One of the people most heavily influenced by Maududi's ideas for recreating society through Islamic law was Sayyid Qutb.