Ok, we need to get some multiplayer going on with some of these games. There are several with asynchronous multiplayer (we don't have to schedule online time, make a turn whenever you want), and now that that functionality is built into iOS5 we should see even more.
Here are some games we can play. Many have free versions, usually ad supported and a limit on the number of simultaneous games you can be involved in. I'll note which ones.
Words with Friends, Chess with Friends, Hanging with Friends
Scrabble, Chess, and Hangman. All three have free versions that are ad supported.
Carcassonne
This one is highly recommended by many. Sadly no free version and the full game is $10 that never goes on sale.
Reiner Knizia's Samurai
Another good one, and one I already own. Again, no free version but this one is only $5.
Poker Pals
Just came out and its pretty cool. Its kind of like scrabble, but instead of making words you're trying to make poker hands. Free version available.
Disc Drivin'
A turn based racing game. Sorta. You have a disc that you flick around a track, racing other players. Its pretty fun. Free version available.
Ascension: Chronicle of the Godslayer
A deck building game like Dominion, I guess. I have a lite version but it looks like it was removed from the store for some reason. So you have to buy it if you want to play.
Kard Combat
Basically a portable version of Spectormancer, a card game designed by the dude that designed Magic. Free with in app purchase to unlock all the mages/decks.
New World Colony
A board game that seems similar to Catan, but not really. I picked it up ages ago when it was free, but theres no demo or anything now.