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!
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Enclave. They are the remnants of the legitimate government, have the most equipment, troops and are technically in the right, in their attempts to retake the wasteland. I do like the Brotherhood of Steel in the first game, as they were the last remaining bastion of pre-war civilization and technology. However in Fallout 2, 3 and NV, they were either out of the picture, a rag tag group of fanatics or worse. Personally I've always made my own way in the wasteland. I treated those in need well and punished the wicked with plasma justice. Then I flipped it around and became the scourge of humanities remnants. That's the best part of Fallout.