At it's most basic, anyone who to witnesses or "to testifies to the belief in the oneness of God and acceptance of Muhammad as God's prophet. is a Muslim.
As in all religions, there are many different interpretations. It would take a very long time to explain all of them. To simplify, think of the vast disparity in Christianity, such as: snake charmers, Christian Zionists, Unitarians, Trinitarians and all the various versions therein. Look at the Roman Catholic church and all the orders therein.
As I see it the main division in Christianity are between Protestant and Catholic, but it's more than that. It gets to the concept of the Trinity or Unity.
In Islam you have Sunni and Shi'a....which really only has become an issue recently. I know many Muslims who are both, and we didn't care or even question it, until the Iraq war started.
As for the idea that Sufism is a separate order or branch, I would not agree with that, it is an integral part of Islam. Rather that idea is an Orientalist concept that indicates that Islam does not have inner dimensions in itself. Even at that, people think of the Mevlevis as the Sufis, but there there are many different interpretations as well. At it's most basic, the word Sufi comes from the word Tasawwuf, which comes from the word purity. Thus those who seek purity are followers of tasawwuf....which became Sufi in the west.
We have seen the spread of the Salafist doctrines from Saudi Arabia growing and spreading in the last 40 years, primarily due to their oil wealth. The Salafis are like Christian "Fundamentalists".
You have Sunni & Shia, in that you have in the Sunni the Hanafis, Malekis, Shayfis, Hanbalis, and in each of these are different schools of jurisprudence, and in them things drill down even more.
In Shia leadership goes through hereditary lineage. It gets complicated....and I don't feel qualified to write a lot about it, but I hope this helps somewhat.
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