Some board games I've played recently:
7 Wonders is a card game where you buy various things to give you production and value in later rounds. I've played quite a few games of 7 Wonders now, and I still feel that I don't fully understand what I'm doing. The scoring is quite complicated. Overall I think it's the weakest game on this list, because it's the hardest to pick up. It works best with at least 3-4 people.
BANG! is a card game based in the wild west. The idea is that one player is the Sheriff, who is drunk and doesn't know who anyone is. The other players are either deputies, outlaws, or the mysterious renegade. Each player, apart from the sheriff, keeps their identity secret, and everyone tries to guess who's side each player is on. The outlaws are trying to kill the sheriff, the deputies are trying to protect the sheriff, and the renegade is trying to be the last man standing. The game unfortunately requires at least 4 players, but it plays best with 6 or 7. I highly recommend it if you can assemble enough people.
Cacassonne is a pretty fun game centring around laying out tiles to build and claim various structures. You want to be claiming structures that will give you points, while obstructing or commandeering your opponents' structures. The rules are simple to learn, but the resulting gameplay is complex and very re-playable. Carcassonne works with more, but the sweet spot is 2-3 players; the more people you have the less you'll be able to do as there are only 72 tiles and the game is over when they have all been placed.
Colt Express is a card game where the cards you play allow you to program a sequence of actions for your character to fulfil at the end of each round. The aim of the game is to steal loot, and have the most loot at the end of the game. The interesting thing about this is that your plans can be disrupted by other players' actions, so when you are planning your sequence of moves, you need to keep track of where you are, where other people might be, and maybe you'll try to keep a contingency plan in mind. It's playable with 2-6 players, but it plays best at the upper end of that range as things get more unpredictable and chaotic that way. It's not really great with two players, but gets a lot more interesting with 3+.
Dominion is a deck building card game where the aim is to buy victory cards, giving you points at the end of the game. I particularly like Dominion for requiring that you have distinct strategies at different points in the game. You probably don't want to be buying victory cards early on, because they'll dilute your hand and hamstring you in subsequent rounds. It works fine with 2+ players.