The Settlers of Catan
Settlers is the first euro-style board game that I ever played. Rules were pretty easy to pickup. Definitely has some element of luck since you are subject to random chance of which spaces are available to you at the beginning of the game as well as the roll of the dice during each turn. One unique thing is that psychology and group dynamics can have large affects through the ability to trade resources. Winning strategies mitigate the feast and famine of the rolls of the dice by diversifying the dice rolls that yield resources. It isn't my favorite board game probably because I lose at a greater rate than other games. I also hate it when the robber gets put on...
I still need to play this one! It sounds fun but I just haven't been in a position to play it!
I too have heard so much about this game and yet I still have never played. Seems pretty rad.
@jdodson and @Adym both need to try this game out sometime so you at least know what people are referring to! Let me know if you need to borrow my game board.