>>716126464Dissidia 012 is the only one worth playing. It's a straight upgrade of the first game (it includes the entire story mode of the first, I think the only thing it cuts is the duel colosseum mode). Dissidia NT is a different genre entirely, made by a different dev, and is terrible.
012, then, is a 3D arena fighter with absolutely shitloads of single player content. Probably 100+ hours of story mode (if you like Kingdom Hearts you'll be right at home, it's crammed full of Nomura nonsense, and not in a bad way). Good roster/playstyle variety, characters to level up and gear, a rock hard endgame campaign with a whole open world to explore, tons of fanservice, runs perfectly on emulator. It's my personal pick for the best game on the PSP, and I think everyone else would put it in the top 5 at the very least. Bear in mind the demo is actually the prologue to the game with a story segment that isn't in the full game, so you'll want to play that as well.