Growing up playing North American games (Monopoly, Life...) I was pleased last year to learn about a fun game originating from Germany called Carcassonne.
Unlike traditional board games, the "board" is made up of tiles that you place in turn, creating a different map every time the game is played. According to Wikipedia, along with introducing wooden little characters called "meeples" to the North American audience, the game has won the "Spiel des Jahres (German for Game of the Year) and the Deutscher Spiele Preis (German Game Prize) awards in 2001".
There are 72 tiles in the original game (there are many expansions) and 2-5 players can score points based on placing one of their seven meeples on the tiles they play. For example, one way of scoring points is to place a meeple on a tile with a castle. You can build bigger castles by adding castle pieces but of course other players can make it hard for you by adding pieces that "don't match" what you need. A bit hard to explain but it takes only a few games to understand it and it is really worth the fun.
Pros: Different game every time. Does not involve as much luck as most board games. I prefer 1vs1 as it can be a bit long waiting for everyone to play.
Cons: In multiplayer (more than 2) mode, it is too easy for collusion to occur, making it easy for a group to determine the winner.