What you’re asking is pretty complicated. Islam was, early on, divided into two main groups, Sunni and Shi’ite, over the question of who would lead the Muslim community after Muhammed died. Shi’ites thought that the Muslims should be led by a member of Muhammed’s family, specifically in that case, Ali, who Muhammed’s cousin and son in law, and one of the first converts to Islam. Sunnis thought that the new leader of the Muslim community should be picked by the community. As of now, about 90% of Muslims are Sunni and about 10% Shi’ite. Shi’a Islam is hierarchical to an extent, but Sunni Islam is not really hierarchical at all.
Sunnis are generally divided into four schools. At the end of the 8th and beginning of the 9th centuries, when the original Muslim community was dying off, Muslims realized they had better write down Muslim law, which is called the Shari’ah. Four different people did this, and they all interpreted it slightly differently, and these differences became the four schools of jurisprudence (Hanafi, Maliki, Shafi’i, and Hanbali.) These aren’t sects, they’re just different ways of interpreting Islamic law.
Now, we need to jump ahead for a little bit. In the 13th century, there was a Hanbali scholar named Ibn Taymiyya. The late 13th century was a bad time for the Muslim world, largely because of the Mongols, who had come in and were in the process of violently overthrowing the Muslim states of the Middle East, and then converting to Islam themselves, and ruling Muslim lands. So this led to a lot of turmoil. Ibn Taymiyya said, basically, that Islam had gotten corrupt, and that it needed to go back to the “original” Islam. . . that Muslims needed to live the way Muhammed and his original followers did.; Specifically, that they needed to get rid of all the practices that had developed since then, like praying to Muslim saints and holy men and the building of and worship at shrines. Ibn Taymiyya also came up with the idea that there could be people who called themselves Muslims who weren’t “true Muslims” (as defined by him), like the Mongols, for instance, and that it was a moral duty to wage war against these people, who were perverting true Islam. Ibn Tamiyya wasn’t popular. The Muslim states were already fighting the Mongols, and ibn Tamiyya’s “Nobody’s a real Muslim except me, and all the rest of you who think you’re Muslim are really just a bunch of superstitious fools” attitude didn’t win him many friends, and he spent most of his life locked up.
So, now, we jump ahead again to 18th century Arabia, where we meet a man, another Islamic scholar, named Muhammed al Wahhab. Al Wahhab “rediscovers” ibn Tamiyya, and decides that things haven’t really changed that much. Islam is still in need of reform, there are still no “real” Muslim states, and the world has pretty much gone to hell ever since Muhammed and the original Muslims died. So he starts a religious reformist movement, and unlike ibn Tamiyya, who nobody listened to, he’s lucky enough to meet and win over a tribal chieftain named Muhammed ibn Saud. Between the two of them, with the resources of ibn Saud, and Wahhab’s fanatical followers, they’re able to take over most of the Arabian peninsula, and the Saudis declare a holy, “true” Muslim state. This gets the attention of the Ottoman Empire, which nominally rules Arabia, and they send an army in and wipe it out. This doesn’t, however, wipe out either the house of Saud or Wahabi Islam, both of which will, about a hundred and some years later, come back in a big way.
Not too long after, in the middle of the 19th century, in India, a madrassah (a Muslim religious school/seminary) is founded in Deoband, and the scholars of this madrassah come to some of the same ideas as ibn Wahhab (although there are some major differences).
These two groups, the Wahabis and the Deobandis, make up the majority of the Salafi movement in Islam, and most of the violent radicals in Sunni Islam identify themselves as Salafi, although most Salafi are themselves not violent. The Rushdie thing was a separate thing. That was primarily the Ayatollah Khomeini (who was a Shi’ite religious leader), who took offense at Rushdie’s book “The Satanic Verses”, even though he probably had never read it, and issued a “fatwa” (a legal opinion) saying that the book was blasphemous, and that Muslims should do their duty and kill Rushdie and the book’s publishers.