I didn't read everyone else's replies, but I did see that someone mentioned Carcassone, and I have to second that. My son is 4 and loves to play Carcassone. Obviously, he's not as competitive as, say, his mother, but he enjoys it and we help him along. The nice thing is, if he gets bored with it (it can sometimes be a long game--especially if you play with expansion packs and, really, why bother if you aren't going to pull out a couple expansion packs?!) he can just drop out without affecting the rest of the game. We can still finish the game if he decides, like he did last night, that he'd rather quit and go design his *own* board game with paper and oil pastels.
Where more kid-centered games are concerned, we're on a HUGE Guess Who? kick right now, and my 4-year-old is loving it. Even my 2-year-old loves playing along and just asking the questions that her dad or I prompt her to ask.
We just taught our son Connect Four, which is also a good one.
Memory is always fun, too. We really like that.
The best "kid game" we own, though, is a Haba game my son got for Christmas last year. It's called Tier Auf Tier, and it's a stacking game with little wooden animals. We love it. We also like Socken Zocken--another Haba game--but we haven't gotten into it as much as Tier Auf Tier.