Fighters are some of the best games to play with others, and Soul Calibur and Dead or Alive 2 are can't-miss. And you can't do much better than Marvel Vs. Capcom 2 (Fifty-six characters to choose from!) when it comes to 2D fighters. Also, if you have multiple controllers, try Power Stone 1 or 2- really fun with company.
Aside from fighters, I second Shenmue and the Crazy Taxi series, and also suggest Space Channel 5. Very fun, and dirt cheap.
