My husband and I have played a lot of computer games together. We also have kids and like to play something casually for an hour or so after they have gone to bed. I would advise you to avoid 1st person shooters, real time strategy or anything that pits you against each other. I learned very quickly that I didn't like being "attacked" in games by my husband.
One of the first games we played was The Longest Journey. We sat next to each other and took turns controlling the mouse. It has a great plot and is an adventure / puzzle game, which might be a good introductory game.
Another really fun one for me was Terraria, which is sort of like a building game crossed with a side-scroller that allows for multiplayer.
My husband and I have been playing a lot of Torchlight II lately, which is an rpg that can be played cooperatively. You could also try Diablo 2, which is in the same vein, but I think Torchlight is more approachable.
Lastly, I really liked playing Civ IV cooperatively.