RPG:
I enjoy playing RPGs, although I have to set aside a lot of time if I want to complete them "properly". I enjoy playing the Final Fantasy games, but some of them irritate me with their questionable character development methods. IE - Final Fantasy VIII's notoriously bad levelling-up system, which allowed monsters to level up along with you. I understand the logic for doing this, but it changed a lifelong FF tradition that I used to enjoy. Thank goodness they fixed this "problem" in FFIX. Aside from the FF games, I enjoy the Zelda Series, Grandia (the first one only, mind), the Little Big Adventure series and plenty more. The music also made me love these games. The FF soundtracks are excellent, and the music in Zelda and the LBA games isn't half bad, either.
Sports:
Aside from the excellent WWE: Smackdown series for the Playstation 2, I don't play any sports games. The glory days of NBA Jam on the Genesis/Megadrive are long gone, Football games irritate me, and I cannot stand driving games no matter how much I force myself to tolerate them.
Platform:
For some reason, I absolutely adore platform games. Aside from the obvious ones I'm about to mention (Mario, Sonic, Yoshi, Ratchet 'n' Clank, Jak 'n' Daxter, Banjo Kazooie and *ahem* Croc...), I also love Klonoa 2: Lunatea's Veil on the PS2 and the lesser known Dynamite Headdy on the Genesis/Megadrive. I still love platform games, and it's a shame they don't make many scrolling ones these days. Viewtiful Joe comes pretty close to being a proper platformer, but it falls into the fighting category. Oh, and I hated Mario 64 and Mario Sunshine. Boy, those games sure sucked.
Fighting:
I've only ever enjoyed a few fighting games during my gaming years, and those happen to be Mortal Kombat II/III, Street Fighter II and Soul Calibur. More often than not, though, fighting games bore me to death. Mortal Kombat was something new in it's day, and mastering all the fatality button combinations was a lot of fun. Street Fighter II was hellishly addictive, and again, memorizing the button combinations was part of the fun. Soul Calibur had great music, great stages, a good story mode and wonderful characters. Other fighting games just don't seem to cut it for me...
I don't really play many other types of game. Not enough to warrant a mention here, anyhow.