Kerry, you're pretty much right
The 2 main groups within Islam are the Sunni and the Shia. basically the division was about succession the Shia believed that caliphs should be hereditary whilst the Sunni believed that it should be based on merit.
They had a big battle about it some years after the death of Muhahmed (FWIW the Sunni were led by a woman). These days the Sunni are the majority. There was also a smaller faction called the Kharijites but they have become a tiny minority - there are some in Oman but that's it.