As with Christianity, the differences within Islam are as much about political power and ethnicity as they are about “religious” (theological/ethical) issues.
The major dichotomy is between Sunni and Shi’a. And they are in agreement that after Mohammed’s death, authority as regards Islam was supposed to pass to a khalifah (Caliph), meaning authorized successor. However, the Shi’a believe that this should properly have only gone to lineal descendants of Mohammed through his daughter Fatima and son-in-law Ali – though they accept that Ali gracefully and humbly permitted the first three Caliphs, who were collateral relatives of Mohammed, to precede him in his rightful position (in their eyes). The Sunni, on the other hand, consider the Caliphate elective and not confined to Mohammed’s descendants. (Ironically, neither group actually has a recognized Caliph today – descendants of Ali being scarce on the ground, and the Sunni Caliphate having passed to the Ottoman Turks and having been dissolved by the last Sultan at his abdication in 1922.) There is also a strong divergence between them on how much authority to give to the Hadith, the oral tradition accompanying the Koran, in regulating the legal and moral behavior of Muslims. As noted, the Shi’ites are broken into smaller groups based on disagreements as to who the rightful Caliph at various times properly was.
Another divergence is on the proper intrpretation and application of the Shari’a, Koran- (and Hadith-) based Islamic law. There are five main schools of opinion. Surprisingly, there is very little conflict between them in the sense of arguments between, e.g., Catholic and Protestant, or Episcopalian and Southern Baptist, more the sort of intellectual debate one might find between constitutional lawyers or historians.
Finally, there are some strong sectarian tendencies over how firmly and strongly various aspects of Shari’a should affect members of Islamic communities. The Wahhabis who have strong influence in Saudi-Arabia (having been among the earliest and strongest supporters of Ibn Saud) are an example of a fundamentalist Sunni sect who believe that their interpretation of Shari’a should be binding on the entire community. The Ayatollahs of Iran are an example of a moderate conservative group among the Shi’a. And the divisions are strongly felt, deep, and complex, giving rise to a very arcane and intricate set of sects, whose doctrinal differences are as opaque to outsiders as the views of the Associate Reformed Presbyterians and the Free Will Missionary Baptists are to the average Doper.
The Sufi are a group of Islamic mystics. The Druze are a group partly theologically and partly ethnically divided from other Muslims, strong in Lebanon and Syria. The Kurds and Azeris (Azerbaijanis) are ethnic groups occupying parts of Iraq, Iran, Turkey, and of course Azerbaijanistan. It’s important to remember that while the original Muslims were Arab and most Arabs (but by no means all) are Muslim, there are a lot of Muslims who are not Arab. The five countries with the largest Muslim population, to the best of my information, are Indonesia, Bangladesh, Malaysia, India, and Pakistan – none of them “Arab” countries.