I put the ages as recommended by the BoardGameGeek Community, rather than the manufacturer. If you're playing a lot of board games, though, and have a kid who generally "gets" games, a lot of the 8+ games can be easily played at 6 or 7, too.
Check out Board Game Geek for more information for these games, or The Dice Tower's many reviews. Dice Tower has very nice videos that show the games and give you the general idea of how it's played.
Animal Upon Animal: An adorable Jenga-like game of stacking chunky, wooden animals on top of each other.
Sequence for Kids: A turn-taking game with interesting choices! Players use their cards to try to make four-in-a-row on the board.
Robot Turtles: Youngsters use some very basic programming ideas to navigate their robot turtles through puzzle-mazes.
Spot It!: There's also a Junior version of this game. Fast-paced matching.
Story Cubes: A game? An activity? Roll the dice, look at the pictures, tell stories together.
Enchanted Forest: One million times better than Candyland.
Catan: Junior: Here's a really in-depth review of this game and its strategy that I wrote. Yes, I thought about it that hard.
Carcassonne: I know, I know. Someone's looking at this like I'm crazy. If you simply don't score farmland, this becomes a MUCH simpler game, one that younger kids can play. When they're older, you can put those rules back in easily. It's great!
Labyrinth: A puzzel-y, spatial reasoning game of finding treasures in an ever-shifting maze. Delightful. Challenging for adults and kids.
Sushi Go!: Adorable, and delicious, card-drafting game.
Black Sheep: I've got a review of this one, too.
Dragonwood: Card hand management and push-your-luck, all while battling fearsome forest creatures! I'm not a huge fan of the cover on this one, but the interior art is bright and eye-catching.
Kingdomino. Okay, here I split the manufacturer's recommended age, and the BBG recommendations. The gameplay is elegant and simple -- like a streamlined, shorter version of Carcassonne. But scoring does require some multiplication (usually 2x or 3x, something 4x or 5x if you've played really well), which could be frustrating for younger players.
Alhambra: Buy up buildings for your Alhambra! Requires quickly being able to add sums up to thirteen.
Castle Panic: Everyone plays together as a team to defend their tower from invading monsters. We've played this one so much our board has worn in two.
Forbidden Island: Another co-op. Everyone plays as a team to recover four artifacts before the island sinks.
Forbidden Desert: This one's a bit harder than Forbidden Island, but on a similar theme. Players work together to find the pieces of an airship to escape the desert.
The Adventurers: The Pyramid of Horus: Get the most loot before the pyramid collapses on you. This is probably my favorite push-your-luck game. My kids love this one, people who don't usually play board games like this one, and my board game fanatic friends also love it.
The Hanging Gardens: Spatial-reasoning card-laying to build the best gardens. Really elegant gameplay. I considered not putting it on the list because I know it's out of print, but I hope someone eventually reprints it.
Mondo: Tile-laying world-building goodness.
Monopoly Deal: So. Much. Better. Than actually Monopoly.
Ponte del Diavolo: Abstract 2-player spatial game.
Quixx: Hard choices with dice. I always lose this one. Badly. Still love playing it.
Scattergories: Okay. This one said for older kids, too. We usually make up our own lists on topics that the kids can easily relate to (Things At a Park). Scattergories is nice because we can play even when there are little kids running around -- there's no board to mess up.
Smallworld: It's kinda like Risk. Except better, and with more interesting choices.
Ticket to Ride: Build trains across the country to earn the most points. There are SO many expansions to this. The base game shown here has a map of the USA, though I think my favorite one is Switzerland.
Munchkin Loot Letter: A deduction game that always has us laughing.
Behind the Throne: Great card game that isn't about trick-taking.