Pnx mentioned in the happy thread about putting together a gaming group with a friend, but not knowing what to run. I started putting together a list of unique games in my collection as potential inspiration and decided this deserved its own thread.
I always want to know what the players of an RPG want before I start making recommendations, because saying "I want to play a tabletop rpg" isn't much less generic than saying "I want to play a computer game". It's certainly more generic than saying "I want to play a computer rpg", and that still covers a lot of ground!
If you like pseudo-historical settings and/or are interested in fairly chunky magic systems,
Ars Magica is the system I've run the most over the past five years. By default, it takes place at the start of the 13th century, and has a lot of the notions of group storytelling that White Wolf has made part of its core, but blended in a way I find more attractive; both feel like they're trying to tell a group narrative, but AM has a strong sense of more equal collaboration among players in figuring out what they want their game to be, and players being able to shift roles throughout the campaign.
If you want a generic system that can do many things, with a focus on the skills of the characters, I think this is where
GURPS reigns surpreme. I prefer 3rd ed over 4th ed, but YMMV. If you want a generic system with an emphasis on powers (good for fantasy, sci-fi, and superheroes), I then lean toward
HERO. (One day, I will sit down to take the time to blend these two generic systems together into a cohesive whole that does both powers and skills well!)
If you want a fantasy game of a less western flavor, I am a big fan of
Legend of the Five Rings. It has a very well fleshed out universe with a great deal of history. It has a lot of eastern motifs. It has the range to do both very high and very low fantasy. It has d20 versions if you're into that sort of thing.
Speaking of d20, if you put a gun to my head and made me play a d20 flavor, I'd go with the Song of Ice and Fire d20 variant that was put out 5-10 years ago. It doesn't quite go far enough to smooth out the obscene d20 power curve, in my opinion, but it tries hard, and it provides quite a good amount of new chrome onto the system (like its influence system), all while using an IP that, at the time, wasn't nearly as mainstream-popular as it is now. (I'm saying this is a Hipster RPG: it liked GRRM before he was cool.)
If you're into things that are more tactical and less about the narrative, d20 is the popular choice, but I'd actually hop on the wayback machine and recommend
Birthright, an AD&D splinter that had the standard D&D stuff, but also a strategy level where the characters are major players in the world who generally control important sections of the world, whether guilds, sources of magical power, strongholds, or, at the high end, entire kingdoms. Thus, there are seasonal moves at the strategic level, where, sometimes, the players decide to go off adventuring for a while.
Also easy to turn into blended tactical/strategic/rpg games are
Battletech if you're into Mecha,
Heavy Gear if you want a slightly more realistic giant robot setting, or several different systems and system blends I can think of for space.
FWIW, here's the books I have hard copies of, and can thus provide greater specific insight on:
http://www.librarything.com/catalog/ndkid&tag=RPG&collection=-1