We'll start with the basics. Two board games you need to know: Ticket to Ride and Settlers of Catan.
Ticket to Ride is our gateway game. We teach it to our friends before introducing the harder stuff. You play as competing railroad tycoons claiming railway routes between large cities (American cities, but there's versions for Europe, Germany, and Scandinavia), earning more points for longer routes and connecting distant cities. Gameplay is a constant tension between purchasing routes (before someone else does) or grabbing more cards (so you can purchase said routes). You can learn this game in 5 minutes and still be playing 5 years later (we are).
Settlers is a bit more complicated, but even more addictive. Players compete to settle an unexplored land. They must collect and trade their resources in order to build roads, settlements, and cities before someone else does. With a board that changes every game and multiple expansions, this one hasn't gotten old since we learned it 12 years ago.
But oh my gosh, guys. That's just a cubic meter of ice on the tip of the tip of the iceberg.
Carcassonne. Claim cities, roads, and fields. Bigger cities (etc.) means bigger points. The gameplay is simple (draw a tile, place a tile). The real trick is figuring out how to encourage other players to complete your cities with their tiles.
Agricola. Plow fields, raise animals, renovate your home, all while making sure your family stays fed. Way more fun and complicated than it sounds. A simpler game ("simple" here is a relative term) with similar gameplay is Puerto Rico, wherein you build a colony and plantations in the New World.
Like the party game Mafia? Try Bang! Kill the sheriff (if you're an outlaw) or the outlaws (if you're the sheriff) or everybody (if you're a renegade). But make sure you know who you're shooting at before you pull the trigger; they might be on your side.
Think that's it? We're still WAY above the waterline here. There's like ten games in our cabinet I haven't even mentioned, a bunch more I've played or heard about, and that's not even counting cooperative games!
I should stop before I lose more readers than I have already... But wait! I haven't told you about new versions of Risk yet. Or Lost Cities! Or Citadels! Or Formula De!! Or --
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