We've always been into playing games, long before we had children. I've loved video games for as long as I can remember (anyone else ever play King's Quest?) and have owned many game systems. B and I also got into (mainly European) board games through some friends and we've never looked back. We were pretty devastated when our friends moved back to the States :o( Our boys love games too (not much of a surprise there) and we're looking forward to playing some of the trickier ones with them when they're a little older.
Dominion with us this year (awesome game) and he's really pretty good at it. One of the best things about all of this gaming for kids is that it teaches them a lot about winning and losing and about how to do it graciously. There have been plenty of melt downs and we're still working on it. NanaD tells the story of us playing "unhappy families" when I was a girl as our games of "happy families" would inevitably end in tears.
If you're thinking about buying a new game please avoid monopoly and the game of life and get something the whole family will enjoy for years. Carcassonne is awesome. We've introduced it to so many friends and they've all ended up buying it. It says ages 8 and up but you can even play it with preschoolers (just modify it a bit), and it works really well as a two player game as well.
As far as kids games go the ones we enjoy are... jnr scrabble, connect 4, hey that's my fish, charades, shopping list (for younger kids), and memory. I'd avoid ... monopoly, buckaroo and the game of life (although I enjoyed it as a kid there are much, much better games out there).
On weekends/holidays when we're at home, we have enforced "quiet time" for an hour after lunch. The boys go off to separate rooms and read, nap or play quietly. DaddyB and I usually make a pot of coffee and have a game or two of Dominion or Race for the Galaxy. We all feel better after a bit of time to ourselves.