Totally depends on the age of the kid. We've done Carcassonne, Dominion, Ticket to Ride, Small World, Roll Through the Ages, Master's Gallery, Bohnanza, Bull in a China Shop, and quite a few others with pretty decent success. We've got a couple that don't quite work as well because they're more complex. The favorite seems to be Dominion and its expansions so far. It plays relatively quickly and the short one often skunks the parents so it stays interesting.  Roll Through the Ages works well – dice, choice, trying to avoid negative points, maximizing scoring. It's easy enough to understand and keep moving.

We've played Monopoly/Scrabble/Clue – all Junior editions. That worked well during the younger years, but we're outgrowing it now. (and my wife is rejoicing)

Guess Who still works pretty well, especially the newer one with the gizmos and multiple boards.

I'd love to build up to a longer game such as Civilization or Railways of the World, but know that would take a while.  We don't have the attention span for a multi-hour game – even Dominion can be a challenge at times.

I'll second Qwirkle as a pretty good game. Forbidden Island works well, but remember to include the kid(s) in the choices. Remember that what they really like will often be different than your choices. Watching Melody & Tom Vasel give their favorites on the Dice Tower top 100 really highlights that. Melody chose Candy Land and Tom just looked at her; Melody had chosen it as a favorite because she could play it with her younger sisters (at which Tom relented :) ). It's also neat seeing the game choices evolve as the kids age. Candy Land is replaced with something completely different. Monopoly falls off completely (with some parental encouragement). Scrabble Junior leads to Bananagrams or Perquackey or sometimes full Scrabble. Clue moves to Mystery of the Abbey or similar. It's fun to watch and join in.

Haven't had a chance to try Dixit yet. We tend more towards games that play 3 comfortably. The larger group / party-type games don't work as well for us.

I'll add in a plug for Quarriors. Our kid does Dominion pretty well so playing Quarriors came pretty naturally. Yes, it's dice-based. It plays quickly, sets up pretty easily, and while it has characteristics of deck-building games, is still a dice-game so luck factors into it. We enjoyed ourselves quite a bit and played 3-4 games in a row. I think the random factors really throw my wife a bit and there are quite a few discussions on how the game might be balanced a bit to keep someone from running away with the victory points, but it's still fun (to us).  The recommended ages are really high on the box as my 8-year-old plays with no issues and has won a couple of games handily.