I'll assume you're asking because you have children or younger siblings, so if you're just looking for games you can play that aren't violent, you can ignore some of my warnings.
Rune Factory 2 for the DS. All the fun of Harvest Moon titles, but when the farming gets boring you can go explore and fight monsters, and eventually you can force them to take care of the plants while you fight monsters, craft weapons, and enjoy a decent story.
Seconding Minecraft. Definitely Minecraft. In fact, stop reading this and go play Minecraft.
...
Why are you still reading? Minecraft is like right there.
*ahem*
But seriously, the game is pretty fun, as long as you watch a Let's Play or get acquainted with the basics in some other way, and find a list of recipes for crafting stuff.
Fable 2 for the 360 has a relatively good story, the combat system is simple, and it has a lot of replayability. However, some of the things you can do if you dip into the evil quests and decisions can be a bit much for young children, so it'd definitely be an early teen game. Basically, are you cool with your kid being able to sacrifice his wife to gain a sword that drips blood, or contract an STD from a prostitute? If so, or if you're willing to monitor his gaming to steer him away from the eviler options, then it's a good option. The humor can be a little crass as well, but that could be a pro or con depending on the person.
Trauma Center for the DS has some blood, but considering that it's a game about surgery, that should be expected. It keeps that type of thing to a minimum though, and if your kid likes biology or science, it could be a good choice. However, the game gets very tough, so a younger child might not be able to play the later surgeries, and anyone easily frustrated should stay away.
My World, My Way is a fun DS game where you play as a Princess who goes off on an adventure to prove to the young hero she has a crush on that she's worth his time. To get a feel for the game's attitude, one of the game mechanics consists of her pouting to change everything from difficulty to the world's terrain. Sounds kind of ridiculous but the game is fun, innovative, and inside the generic "defeat the big bad" plot is some great character development.
The Ace Attorney series for the DS has a fun story, and will be a hit if your kid enjoys puzzles and brainteasers. However, you will be investigating murders, so that might not fit your idea of kid-friendly. Plus some of the weird twists of logic in that game require a lot of trial and error or easy access to a strategy guide.
Scribblenauts (DS) wasn't quite what I would call a great game, but the sequel Super Scribblenauts supposedly has fixed its most glaring problems (poor controls and lackluster puzzle design) so once again, it could be a good game for the kid who enjoys puzzles.
I never played the first one, but Patapon 2 (PSP) is a fun rhythm game/RTS. If you don't find yourself bobbing your head to the beat or singing along, you might not have a soul. Just saying.
Speaking of rhythm games, Elite Beat Agents (DS) is incredibly fun and completely ridiculous. If being a trio of federal agents tasked with boosting the morale of various citizens through the power of cheerleading synchronized with some very catchy songs sounds like a fun time, this is your kind of game.
The Legend of Zelda series as a whole may have a lot of violence packed in, but it's almost completely cartoon violence, aside from a few of the more dramatic boss battles. I'd recommend one in particular, but I couldn't single out any as a game I wouldn't of wanted to play as a kid. If forced to choose, I loved the music and generally relaxed/whimsical atmosphere of Windwaker (GC), and the choice of graphics means it will always look great.
Outpost Kaloki X, available on the XBLA, is an enjoyable game of resource management, where your objective in each level is to build up a space station that meets the requirements of whichever NPC is bossing you around at the moment. The difficulty comes in balancing income, power generation, maintenance, and the desires of the customers that frequent your shops, restaurants, and parks while getting everything done within the time limit.
Ok, I'm going to stop typing now, this is long enough.
P.S. Don't inflict Pokemon on your children. You're just dooming them to a life of griping over how the story of the latest game is still exactly the same, complaining that the minor improvements to mechanics are barely enough to merit a .1 increase in version numbering on any self respecting PC game, and arguing with others online about how the 500 that were in the game when they were a kid are so much better than the 4500 they've added since. Which is wrong, as the original 151 are the only ones worth anything.