First, I'll make this clear: I'm not a big gamer. If I'm on a computer, it's more likely to be reading political essays or blog posts than playing a game. I spend much more time talking with my wife than playing games with her. And in social settings, a rousing discussion is typically more appealing to me than a party game. However, I do enjoy games of all sorts, leaning toward those of a strategic or abstract nature.
Growing up, my family played lots of games. My Mom had played tons of games as a kid and introduced many of them to us. Most of them used a regular deck of cards, so I learned classics like Casino, Hearts, Rummy (favorite variation: "Shanghai," an early version of Phase 10), and plenty of others. We couldn't afford many games, so Uno, checkers, chess, Sorry, Monopoly and Sequence were staples. Abstract strategy games Pente, Mancala and Othello were personal favorites. Popular party games were Taboo and Balderdash. Gambling games were generally looked upon unfavorably.
As I grew older, I spent more time with my cousin Isaac, a pretty avid gamer. We played chess (to this day, I have never beat him), Risk (which I now only play to prevent him from winning), Stratego and plenty of others he aquired from thrift shops. He never managed to get me to play Axis & Allies. It wasn't until we were both adults with a little more money to spend that we were both exposed to the larger world of European games.
That said, I get a bit overwhelmed with lots of small pieces and I don't like spending hours poring through a rule book to learn how to play a game. So for me to learn a new strategy game generally requires a friend to introduce it.
As an adult, my favorite discovered games have been Blokus Trigon and Rumis (both unfortunately out of production). Others that I've greatly enjoyed include Acquire, Dominion, Stone Age, and Agricola. I now enjoy an occasional game of poker. I'm not a huge fan of Settlers of Catan as I am horrible with dice and NOT a diplomat (thus tend to lose). I love playing Dutch Blitz on a team with my wife since she is super fast and with her "help" I can often win.