There are many more factions but these are the most "popular" in my opinion..
You can also choose:
-Followers of the Apocalypse
-Fiend
-Boomer
-Brotherhood Outcast
-Tribal
-Republic of Dave
-Member of Oasis
-White Leg
...and many others that I missed!
Make sure to post why you would classify as a part of this faction and why you like/dislike it!
-
I would like to mention that I support [i]Lyon's[/i] sect of the brotherhood. The primary BOS mentality is douchery and non conducive to fixing the wasteland. The NCR are moreover the more balanced faction as they offer fair treatment to it's people, and isn't above working with the other people in the wasteland.