I don't really know of any "board game" or "Party Games" style games, but out of my library, some that might be applicable for family gaming: (Disclaimer: Player count is from memory, haven't played some of these in a while. Disc based games marked with *, all others are downloads from PS Store )
FAT PRINCESS! Cartoon violence, (slightly) mature humor. Lots of fun though, four-local-player online multiplayer. I'd feel comfortable playing with all ages.
DeathSpank, Maybe? There's cartoon violence(running around beating up chickens and evildoers with rediculous weapons) and some mature humor , but (and I haven't played very far) it's pretty tame so far. Might be suitable for older kids. It's a two-player RPG (well, one player and an assistant)
Smash Cars; racing RC cars. Do stunts, race... pretty self explanitory. Expansion adds local multiplayer (2 player I think, might be 4). All ages.
Sports Champions*: Do sports with the move. What you'd expect it to be. Two-player. All ages.
EA NHL 3-on-3 arcade. Play hockey with bobblehead-looking characters and devious powerups. If you've got six controllers, you can play with six people. All ages.
Mushroom Wars: Kind of like a greatly simplified RTS. Expansion adds online and expands the local multiplayer functionality. 2-player or 4-player (i think) with the expansion pack. All ages.
Tank Battles: Cartoon battle tanks blow each other up. Powerups, different maps. Can be fun. Local and online multiplayer. War themes, but again, cartoony enough I'd feel comfortable with all ages.
Turtles In Time: Re-Shelled - An OK reworking of the arcade game. IIRC up to four player co-op campaign. Classic Ninja-turtles beat-em up. Cartoon violence. All ages.
Trash Panic: Rediculous Japanese-made game. Sort of like tetris, really not but thats the closest comparison I can come up with. Strangely enough the vs mode in this is the game of choice when friends want to play PS3.
2-player local. No objectionable content of any kind.
Age Of Booty: Excellent little pirates game. Collect resources, upgrade your ship, capture towns, win the day. Free add-on maps from PS Store. 4-player co-op or competetive. all ages
Calling All Cars: Another excellent game. Pick your cop car and run down criminals to catch them, and get them back to prison before another player bumps you and steals him to take back himself. This game gets intensely heated sometimes. Good fun, but one of my friends is a little too good, and losing constantly gets old
4-player. all ages
Some of these may be too complex for very young kids but apart from
maybe DeathSpank they should all be content-appropriate for most kids, depending on your parental discretion
EDIT - OH forgot Portal2*
+1
