The largest, Sunni, takes what we would consider a more conventional approach to religion in general.
They represent the majority of Muslims in all but two countries, Iran and Iraq. They have a greater trust in secular type governments and take a broader philosophical perspective. I believe that this perspective is based on a broader understanding of the significant contribution that Islam has made (in math and science) and therefore have confidence that over time Islam isn't just a blessing to Muslims but contributes significantly to world understanding.
This important era forms the philosophical basis of the Western Enlightenment but is largely ignored in our schools. You can read about it here:
http://en.wikipedia.org/wiki/Golden_Age_of_Islam
Shias broke away almost immediately after the death of the Prophet over secessionist issues with many of Mohammed's closest relatives, including his daughter and son in law, objecting to the succession of the caliphat by Mohammed's father in law.
Eventually Ali (son in law) and daughter Fatimah take over the 4th Caliphat but Ali is assassinated. Shia's believe that Allah chose Ali and that the succession away is the result of false actions against Allah.
In terms of theology the differences are that Sunni followers separate government and religion into two different spheres and that Allah chooses leaders of both.
Shia believe that a government should be formed like a Caliphat (Theocracy) and that the leaders should lead through a group leadership and no one cleric ever gets too powerful.
The result is that today many Shia consider the Sunnis to be cultural and political 'sell outs' and many Sunnis consider Shia to be fanatical 'end of timers'.
During the middle ages when Christendom was going through tremendous violence and the inquisitions the Islamic countries had the most tolerance, the best libraries and universities. Algebra and the Scientific Method were developed during these times.
The largest Islamic countries in the world are Indonesia, Pakistan and Bengladesh. There are more Muslims in China than Saudi Arabia. So Islam is mostly an Asian religion and most Muslims live in peaceful settings. Indeed Muslims around the world look at the US and wonder why there is so much violence here, especially when they read the constant barrage of spree killing at the work place and at schools.
Sufism is a mystical form of Islam http://en.wikipedia.org/wiki/Sufism that actively encourages their own type of meditation and peace.
The Islamic Institute in Mannheim, Germany, which works towards the integration of Europe and Muslims, sees Sufism as particularly suited for interreligious dialogue and intercultural harmonisation in democratic and pluralist societies; it has described Sufism as a symbol of tolerance and humanism—nondogmatic, flexible and non-violent.[141]
You might be surprised to find that these are well remembered teachings of some of the Sufi prophets
“Prophet Muhammad, the Rasulullah (Sal.), placed the Qur'an in the hands of his followers and said, "In order to understand this, go even unto China to learn divine knowledge ('ilm)." Within one word in the Qur'an, there are thousands of meanings. Don't hold onto to just one meaning. Look within and there will be another meaning. Within that is another meaning, and within that is still another. As you uncover meaning after meaning, you will see Allah at the very end. When you go beyond all the messengers (rasuls) and look within, you will find Allah. The angels are also there. Go beyond them and look, go beyond everything and look, and then you will see heaven. Go beyond heaven, and Allah is there. Do not pray for heaven. Go beyond and look. Allah will be there."
Before we try to destroy someone else, we should first pass judgment on ourselves. Before finding fault with others, we must first pass judgment upon ourselves. Before we backbite others, we must first pass judgment upon our selves.
Faith is a restraint against all violence, let no Mu’min commit violence.
If you look at organized political violence, whether it is the Bolsheviks, the Nazis or Al Queda it is interesting to note that those that seem to be most ripe for violent revolutionary action are those from an aspiring middle class that were then faced with little or no professional or occupational options. People like Mohammed Atta (and engineering graduate in Egypt who went on to study in Germany) get their expectations raised that they have done everything right. In reality economic development in many of these countries left many of their best students without reasonable chances of employment, in part because nepotism keeps the few good jobs reserved for family members of the elite.
Most Muslims would find your question surprising because they view their countries as peaceful and the US as violent.
For example the murder rate in the US is 4.7 per 100,000 while the murder rate in Saudia Arabia is 1.0, Egypt 1.2, Tunisia 1.1 although a strict statistical comparison is difficult because our statistics are more likely to reflect a more complete number.
http://en.wikipedia.org/wiki/List_of_countries_by_intentional_homicide_rate
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