We've been having a lot of fun with Codenames and Exploding Kittens.
Not your typical Monopoly or Chutes and Ladders.
Real games. It seems a large number of the board games we really enjoy come from Germany. We asked a German friend about this and his theory was that many adults there still play board games on a regular basis. Rather than watch TV, as people do here, they get together and play games. Mark and I particularly enjoy Klaus Teuber's game design (Settlers of Catan). There are thousands of games out there, so it's hard to know what has staying power. These are some of our favorites.
There's the basic version, and then seemingly hundreds of add-on modules. We bought a few of those and enjoy them. However, recently we played with a friend who hadn't bothered with any of the new modules, and gameplay still holds up at its simplest.
Settlers of Catan
Very simple, but with many possible strategies to win. And it's a beautiful game.
Ticket to Ride
Build your railroad! Faster and farther! Very fun. Lots of little train pieces. (You notice that more with a toddler around.)
This is a cooperative game which can be very difficult to win, but is always entertaining to play.
Not as much a classic as the other two, but clever and cute. It's designed for children, but rather than dumb down the game play, it instead plays over a shorter time (15 minutes rather than hours, for shorter attention spans). We discovered this little gem at Gencon and approve of this company's focus toward bringing stimulating activities to families with children that everyone can participate in (again, rather than more TV time).
This is a company rather than a specific game. Their theory is that most games are so expensive because of the high production values and the myriad redundant figurines and counters. They just print up the rules, cards, and boards on plain paper and sell it in an envelope, relying on fun concepts and game play to capture their audience. Fun and cheap, they frequently are very clever and very funny. Two of our favorites are 'Gimme the Brain', and 'Kill Doctor Lucky'.
There are many, many systems out there, and I may be in danger of losing my gamer geek card, but I honestly don't think it matters which you use. We use Dungeons & Dragons, but it's less about the system and more about playing creatively with people you enjoy. You can invest hundreds in the support materials, or you can just buy one player or GM guide and use your own imagination. I prefer the latter.
We've helped GM for this group's three-round tournament for years at Gencon. If you're looking for something madcap and fun, more story- rather than rules-based, you need to try out this tournament.