dude for fighting definitely one of the Tekken series for PS/PS2. I only played up to Tekken 3 like 6 yrs ago so dunno anything about tekken 4. But #3 was awesome, definitely has to be one of the best fighting games (and I see it all over in arcades, so it's quite popular, it seems).
As for "other", u HAVE GOT to have a strategy game. An RTS probably, since turn-based ones take tooo long to play multiplayer (never done it but can imagine). For example Starcraft, warcraft (3), C&C Generals, hmm possibly one of the Age of Empires series? SC, WC, and C&C G. are probly the best choices though.
Also, for "other" u could include a combat flight sim. IL-2 Sturmovik offers multiplayer face-offs online over ubi.com (or so I read) ... still haven't tried it but i want to some time.
RPG is just not gonna work for this imo lol. Any multiplayer RPG game is too long (like any one of the MMORPG's)
Finally, for FPS battlefield 1942 is also really really sweet (haven't tried BF2). Although I'm probably kind of biased cuz I love WW2-themed games...