Games I own:
Dominion (except for Alchemy and Prosperity): Obviously awesome.
Twilight Struggle: It is really good, but it's hard to find time and someone to play against.
Puerto Rico: This was probably the first board game I got really into. In university my friends would play this game constantly.
Carcassonne: It's good, but it usually comes down to the farmers. I've heard some of the expansions deal with my complaints, but I've never investigated.
Settlers of Cataan: I almost never play it anymore.
Brass: It's a little long, but it's actually a really well designed game. It's so dedicated to accurately representing this portion of history and in spite of that it's still fun and balanced.
Kill Dr. Lucky: Basically a prequel to Clue. I have the old Cheap Ass version. It's okay for a few rounds, it's certainly funny, but it gets a little repetitive once people get a hang of the strategy.
Blokus Trigon: I prefer this to the original Blokus. I think the geometry of triangles makes it better. It's fun, but not everyone gets into this super abstract type of game.
Pandemic: I prefer, but do not have the space for Arkham Horror, but this is okay if you want to play something cooperative. Suffers from the fact that you could play it by yourself.
Innovation (with expansion): Innovation is fun, but incredibly swingy. The expansion makes an already complicated game much more complex. I wouldn't bother with it, but the original is neat and I love the physical design of the game.
Aquaretto: If you were put off by Zooloretto, this game takes all the problems of Zooloretto and fixes them. It's a drafting game combined with a space management game combined with running an aquarium.
Chess, Backgammon, Crib: My favourite of the classics.
Pente: A simple, but deep game played on a Go board.
Galaxy Trucker: Crazy and fun. Really swingy, but it's so chaotic and the theme is so fun, that I never seem to mind.
Scrabble: I don't play it much, but I do like this game. It's a classic.
Bananagrams: All the fun word knowledge of Scrabble with the space control game removed. There's greatly accelerated gameplay. I would recommend this to anyone.
Dixit: A good game for non gamers. Pick it up if you're looking for an alternative to Apples to Apples.
Power Grid: I'm sure you've heard of it. Drafting, auctioning, resource management, expansion. It's got a lot of great things. It's better with more people, but my biggest complaint would be that rules are finicky. I have never played it without having to dig through its rule book.
Stuff I'd like to add:
-Eclipse
-Cosmic Encounter (thanks Axxle!)
-Nefarious
-Ticket to Ride