This might be one out of multiple entries. It was hard out of 30 or so years of relevant games to pick only one favorite for each genre, so I tried to get some variation in terms of modernity/sub-genre/etc. These are just a few of my favorite games:

Favorite RPG: Skies of Arcadia
This game was fun for a few major reasons. One, the weird steampunk-ish air pirate world was well-realized and fun to dive into. Two, I loved how most characters were typically optimistic and driven, which created a vast number of memorable, likeable main/supporting characters. Three, it was the first time a game made it fun for me to explore. The discovery system was rewarding both in terms of fleshing out the world, and for rewarding you with money (encouraging more exploration). The battle system is about as basic as it gets, but it's a solid JRPG that sucked me in with a unique, enjoyable style.

Favorite Action game: Metal Slug
I always just liked the first game stylistically more than the others. There were no real gimmicks like zombie-/mummification, and the difficulty was set just right so that it was difficult until you could get used to how to deal with the different combinations of enemies.

Favorite platformer: Super Meat Boy/Psychonauts
I know people will cry foul for not mentioning one of the old classics, but Meat Boy is just a god-damned blast. The developers didn't include a lives system, which forced them to focus solely on the level design to deliver difficulty, which they did really well. The art style, music, and tons of extras also added immeasurably to the enjoyment.

On the other hand, Psychonauts is a game with some dodgy gameplay, but it does so much else right (humor, writing, level design, puzzles, boss battles, etc.) that it's easy to forgive a few moments of "but I landed right on the damned ledge!"

Favorite FPS (god, this list was a lot longer at one point): Painkiller/Brothers in Arms 2/Half Life 2
Painkiller just felt like the modern equivalent of the '90s shooters. Using kick-ass awesome weaponry against legions of cool-looking enemies in gigantic environments was just violent, over-the-top fun.

Meanwhile, Brothers in Arms is one of the few strategy-ish series I like. It sets out some basic rules, and lets you figure out different ways of capitalizing on the mechanics. BiA2 just stands out for having more difficulty, more strategic possibilities, and more extra features than the other two (never played the handhelds). The series also stands out as being one of the only well-written war games, with actual, fleshed-out characters that you hate to see die as it goes on.

Finally, Half Life 2 is just fun. The combat is fun, the characters are all likeable, and everything just feels real and right. I love when games create awesome stories without having to frequently take control or imagination away from the player. Sometimes it's nice to leave a player in the dark, and create some intrigue.

Illbleed was awesome, and I'll put it in my "unknown" pile. I hated it when I first bought it (used, no instruction manual), but read up on it a little and finished a year or two later. It's basically the video game version of Creepshow.

The Specialists was a fucking blast.