Some ideas to look into:
Ticket to Ride (2-5 players) Full honesty, this game is not my thing, but I know tons of people who love it. You draw cards to send trains to various cities.
Settlers of Catan (2-4 players, or up to 6 if you buy the expansion) Family friendly strategy game where you collect resources to build cities and roads. My family loves this one.
King of Tokyo (2-6 players) A king-of-the hill style game where everyone plays a giant monster attacking Tokyo.
The Resistance (5-10 players) Good if you have a big family that likes arguing and getting into games. Some players are secretly spies, and the resistance members try to figure out who they are.
Munchkin (3-8 players) Card game involving lots of backstabbing of other players. You're trying to reach level 10 by fighting ridiculous monsters and interfering with opponents. Lots of different themed versions to choose from (pirates, zombies, kung fu, western, etc).