I’ve got a six-year-old daughter, and we like to play lots of games.
Lego Creator–a board game where you go around the board and collect pieces to create the Lego model on your card. Some spaces let you steal bricks from other players, so it gets quite interesting then :).
Aquarius–a card game published by Looney Labs. It’s sort of like hippie-themed dominoes with cards. You match the pictures of the different elements (earth, air, water, fire, and ether), and when you get a group of seven of your element together, you win. Of course, what often happens is that you get a big group together, and then someone plays a card that changes your element!
Monopoly Junior Dig’n Dinos–It’s like Monopoly, except you are a paleontologist who is going around the board buying dinosaur fossils (and charging other players for the privilege of seeing your fossil when they land on it). Much, much shorter than a game of regular Monopoly (which my kid likes too, but I get bored).
Checkers and Chinese Checkers–Good strategy games that are easier to learn than chess. My daughter likes Chinese Checkers a lot.
Backgammon–My husband has started to teach her this game (I don’t know how to play). His ex-roommate is from Iran, and apparently backgammon is very popular there. It appears to have a good mix of luck and strategy.
Go Fish–The only problem with this game is that it’s sometimes difficult for kids to keep all those cards in their hand!
Uno–Don’t bother with the “kids” version, just get the regular one. A new set even has symbols for “reverse” and “skip”, so that kids don’t even have to remember what “R” and “S” stand for. (My kid doesn’t read well yet, but she knows the cards by sight.)
Uno Stacko–(aka Jenga with colored bricks) You roll the die, and it tells you which color of brick you can pull out. Kids like it when the tower crashes!
Hope this gives you a good start!