The subject of Islamic extremism can be difficult to  grasp.  And, often times, the truth is  difficult to accept.  So, where does one  begin to gain knowledge?  - It is a best practice to start with definition.  Make it a point not to miss the section titled "
MUST READ".
Muslim : an adherent of Islam
Islam 1 : the religious faith of Muslims including belief in 
Allah as the sole deity and in 
Muhammad as his prophet
Islam 2 a : the civilization erected upon Islamic  faith
Islam 2 b : the group of modern nations in which  Islam is the dominant religion
Islamic  Faith : aka Islam (from Wikipedia) : is the 
monotheistic religion articulated by the 
Quran, a 
text considered by  its adherents to be the verbatim word of God, and by the 
Prophet of Islam Muhammad's teachings and normative example (which is called the 
Sunnah in Arabic, and  demonstrated in collections of 
Hadith).  Islam literally means "
submission (to God)." … Muslims believe that 
God is 
one and incomparable and that  the 
purpose of life is to 
worship God. They regard their religion as the completed and universal version of a  primordial, monotheistic faith revealed at many times and places before,  including, notably, to the 
prophets Abraham, 
Moses and 
Jesus. Islamic  tradition holds that previous messages and revelations have been 
changed and distorted over time.  Religious practices include the 
Five Pillars of Islam,  which are five obligatory acts of worship. 
Islamic law touches on virtually  every aspect of life and society, encompassing everything from 
banking and 
warfare to 
welfare and the 
environment.
Islamize : to make Islamic; 
especially : to convert to Islam
Islamization (from Wikipedia) : the process of a society's conversion to the religion of  Islam.
Extremist 1 : the quality or state of being 
extreme
Extremist 2 : advocacy of extreme measures or views : 
radicalism
Radical 2 : of or relating to the origin : 
fundamental
Radical 3 a : marked by a considerable departure from  the usual or traditional : extreme
Radical 3 b : tending or disposed to make extreme  changes in existing views, habits, conditions, or institutions 
Radical 3 c : of, relating to, or constituting a  political group associated with views, practices, and policies of extreme  change 
Radical 3 d : advocating extreme measures to retain  or restore a political state of affairs
Islamic  Extremism (from Wikipedia) : is a term used to describe the actions and  beliefs of 
violent Islamic activists, including 
terrorism and  inhibition of freedom of speech.
Islamic  Fundamentalism (from Wikipedia) : is a term used to describe religious  ideologies seen as advocating a return to the "fundamentals" of 
Islam: the 
Quran and the 
Sunnah.
Islamic Fundamentalism seeks
 “to re-establish Islam as a  system of government… and 'the holy warriors'… are called upon to undertake the  reinstatement of Islam in both its doctrinal and political dimensions. The  method of struggle is often referred to as jihad, or holy war launched in the  path of God.” [See 
Reference 1]
Jihad 1 : a holy war waged on behalf of Islam as a religious duty; also : a  personal struggle in devotion to Islam especially involving spiritual  discipline
Jihad 2 : a crusade for a principle or belief
Holy War : a 
war or violent  campaign waged by religious partisans to propagate or defend their faith
Sharia : Islamic law based on the Koran
Jihadist : a  Muslim who advocates or participates in a jihad
Facism 1 : a  political philosophy, movement, or regime (as that of the Fascisti) that exalts  nation and often race above the individual and that stands for a centralized  autocratic government headed by a dictatorial leader, severe economic and  social regimentation, and forcible suppression of opposition
Facism 2 : a tendency toward or actual exercise of  strong autocratic or dictatorial control
Islamofacism (from Wikipedia) : is a 
neologism which draws an analogy between the ideological characteristics of specific 
Islamist movements from the  turn of the twenty-first century on, and a broad range of European 
fascist movements of the early twentieth century, 
neofascist movements, or 
totalitarianism.
God  in Islam (from Wikipedia) : known in Arabic as 
Allah, is the all-powerful and  all-knowing Creator, Sustainer, Ordainer, and Judge of the universe. Islam puts  a heavy emphasis on the conceptualization of God as strictly singular. God is  unique and inherently one, all-merciful and omnipotent. According to tradition  there are 
99 Names of  God each of which evoke a distinct attribute of God. All these names refer  to Allah, the supreme and all-comprehensive divine name. Among the 99 names of  God, the most famous and most frequent of these names are "the  Compassionate" and "the Merciful".
God (from Wikipedia) : is the English name given to the singular 
omnipotent being in 
theistic and 
deistic religions (and other 
belief systems) who is either  the sole 
deity in 
monotheism, or a single  deity 
in polytheism. 
God  in Christianity (from Wikipedia) : is the 
eternal being that created and  preserves the universe. The 
Christian  Bible never speaks of God in an impersonal sense. Instead, it refers to him  in 
personal terms — as  one who is, who speaks, who sees, hears, acts, and loves. God is understood to  have a will and personality and is an 
all powerful, divine and  benevolent being. He is represented in 
Scripture as being primarily  concerned with people and their 
salvation.
Christianity 1 : the religion  derived from Jesus Christ, based on the Bible as sacred scripture, and  professed by Eastern, Roman Catholic, and Protestant bodies
Christianity 2 : conformity to the  Christian religion
Christianity 3 : the practice of  Christianity
Christian 1a : one who professes belief in the  teachings of Jesus Christ
Christian 2a : (1) : disciple (2) : a member of one  of the Churches of Christ separating from the Disciples of Christ in 1906 (3) :  a member of the Christian denomination having part in the union of the United  Church of Christ concluded in 1961
Disciple 1 : one who accepts and assists in spreading the doctrines of  another: as 
a : one of the twelve in the inner  circle of Christ's followers according to the Gospel 
accounts b : a convinced adherent of a  school or individual
Disciple 2 : capitalized : a  member of the Disciples of Christ founded in the United States in 1809 that  holds the Bible alone to be the rule of faith and practice, usually baptizes by  immersion, and has a congregational polity
Faith  1 a : allegiance to duty or a person : 
loyalty b : (1) : fidelity to one's promises (2) : sincerity of intentions
Faith 2 a : (1) : belief and trust in and loyalty to  God (2) : belief in the traditional doctrines of a religion 
b : (1) :  firm belief in something for which there is no proof (2) : complete trust
Faith 3 : something that is believed especially with  strong conviction; especially : a system of religious beliefs
Christian  Faith : aka Christianity (from Wikipedia) : is a 
monotheistic religion based on the life and  teachings 
of Jesus of Nazareth as presented in the 
New  Testament. Christianity comprises three major branches: 
Catholicism, 
Eastern Orthodoxy (which parted ways with Catholicism in 
1054 A.D.) and 
Protestantism (which came  into existence during the 
Protestant  Reformation of the 16th century).   Protestantism is further divided into smaller groups called 
denominations.
The Four  Spiritual Laws of Christianity:
Law 1 : God loves you and offers a wonderful plan for your life.  
Law 2 : Man is sinful and separated from God. 
Law 3 : Jesus Christ is God’s provision for man’s sin. God has bridged the gulf that  separates us from Him by sending His Son, Jesus Christ, to die on the cross in  our place to pay the penalty for our sins.
Law 4 : We must individually receive Jesus Christ as Savior and Lord; then we can know  and experience God’s love and plan for our lives.  When we receive Christ , we experience a new birth. 
The Four Spiritual Laws flash version can be viewed 
HERE.
Jesus  in Islam (from Wikipedia) : The Quran, considered by Muslims to be  God's final and authoritative revelation to humankind, mentions Jesus  twenty-five times. It states that Jesus was born to Mary as the result of  virginal conception, a miraculous event which occurred by the decree of God. To  aid in his ministry to the Jewish people, Jesus was given the ability to  perform miracles, all by the permission of God rather than his own power.  According to Islamic texts, Jesus was neither killed nor crucified, but rather  he was raised alive up to heaven. The Quran states that he will return to Earth  near the day of judgment to restore justice and defeat "the false  messiah", also known as the Antichrist.
MUST READ : The  Quran and the Four Spiritual Laws – A paper by Pastor Shahrokh Afshar
Additional References:
1.  Choueiri,  Youssef. 
Islamic fundamentalism Copyright © 1998, Routledge.  
Islamic Philosophy From  the Routledge Encyclopedia of Philosophy, Liverpool John Moores University, UK.  Date of access 8/12/2010 
http://www.muslimphilosophy.com/ip/rep/H007