My favorites:
Best PS2 Exclusives (vs. XBox):
-Champions of Norrath (Diablo-Clone, good one)
-Champions: Return to Arms (sequel to above game)
-Contra: Shattered Soldier (Run-N-Gun)
-Drakan: The Ancient Gates (Action-Adventure with RPG-elements, and Flight-Combat on a Dragon)
-God of War 1 & 2 (Action-Adventure)
-Gradius III & IV (Shoot-Em Up, both on one release)
-Gradius V (Shoot-Em Up)
-Guitar Hero Series (Music/Rhythm)
-King's Field: The Ancient City (First-Person Action-RPG)
-Mega Man X Collection (Mega Man X 1-6)
-Raiden III (Shoot-Em Up)
-Red Star (Beat-Em Up & Run-N-Gun (depending on your play-style))
-Resident Evil 4 (Survival-Horror)
-Resident Evil: Code Veronica X (Survivial-Horror)
-Rygar: The Legendary Adventure (Action-Adventure)
-Sega Genesis Collection (Classic Game Compilation)
-SNK Arcade Classics Vol. 1 (Classic Game Compilation, Neo-Geo Games)
-Twisted Metal: Black (Vehicular Combat)
-WWE: Smackdown Series (Wrestling)
Good XBox Exclusives (vs. PS2):
-Arx Fatalis (Action-RPG)
-Crimson Skies (Flight-Combat/Adventure)
-Dungeons & Dragons: Heroes (Diablo-Clone, good one)
-The Elder Scrolls III: Morrowind - Game of the Year Edition (Open-World Action-RPG (huge and detailed), get the GOTY edition, it includes all PC expansions)
-Halo 1 & 2 (Sci-Fi FPS)
-Jade Empire (Action-RPG)
-MechAssault 1 & 2 (Action/Mech-Combat)
Great Games on Both Systems:
-Baldur's Gate: Dark Alliance 1 & 2 (Diablo-Clones, very good for a console version)
-Capcom Classics Vol. 1 & Vol. 2 (Classic Game Compilations)
-Dynasty Warriors Series (Action with a very-light sprinkling of Strategy)
-The Incredible Hulk: Ultimate Destruction (Action, with LOTS of destruction)
-Marvel: Ultimate Alliance (Action-RPG)
-Mega Man Anniversary Collection (Classic Game Compilation, MM 1-8 + Unlockable Games)
-Pac-Man World 2 & 3 (3D Platformer)
-Red Faction II (Sci-Fi FPS)
-Sonic Mega Collection Plus (Classic Sonic Game Compilation)
-Soul Calibur II (3D Fighter)
-Spider-Man: The Movie (Action)
-Star Wars: Battlefront 1 & 2 (Third-Person Shooter/Action, but mass chaos and Star Wars-ey)
-Street Fighter Anniversary Collection (Classic Game Compilation, all SF II variations & SF III)
-Taito Legends 2 (Classic Game Compilation, heavy on the Shoot-Em Ups)
-Tony Hawk Series (Skateboarding)
-X-Men Legends 1 & 2 (Action-RPG)