We play games with our 7 (about to be 8) year old and he loves the following (some have been mentioned and some were over the recommended age for him when we started playing so we had to help him out but within a game or to he was playing without help);
Catan Junior - as was previously mentioned, this version is simplified Catan but still retains a lot of what makes Catan great. It's pirate themed and plays in about a half an hour. He mastered this in one game.
Carcasonne - Simple tile placement game. Easy to learn, games are always different due to the map building element and plays in about an hour. We have a few mini-expansions for it too. He still needs a little help planning ahead but he loves build big cities.
Smallworld- Great theme elements and the variety of armies that you get mean that the games are always different. The rules for this one took him a little longer but he loves the game.
Forbidden Island(Desert) - This is one of his favorites because we all play against the game as a team and he loves the we all help each other. He has a pretty good handle on F. Island so we recently got F. Desert which is a tougher game (still haven't beaten it but he loves the process of both.
Those are just a few of the games we started him on. He has since played Agricola and Caverna and while he hasn't won a game of either yet he loves building his own placemat farm/dwelling.