Great thread. I enjoy chess as well but I mostly prefer to play with live player. Just want to throw it out there, since someone pointed out that playing real-life players is the way to improvement - there are online game rooms where you can play chess behind your curtain
I'm not the most knowledgeable when it comes to all things chess (engines, AI, etc.), but Deep Green is great, and extremely well crafted.
Stockfish Chess (originally known as Glaurung) is, without doubt, the best free Chess app. But for online play, you could give the Chess.com app a try.
I've developed a (non-chess) app and also play a bit of chess myself and IMO although much that's been said on here is true, playing against a computer CAN improve one's chess. Computers tend to be especially good at tactical play (although they're now just very good at everything) and for observing tactical patters playing against a computer is very useful. It's also useful for seeing how to defend well. For example, a human might make a Greek Gift sacrifice to you in human play and it might work and he/she wins. But such sacrifices rarelywork against machines and watching how they defend against is (to me) fascinating. Also, I'd recommend Chess Genius. It's not as strong as Shredder and Fritz (but really, who cares about that other than a few GMs?) but it's probably the most user-friendly, easiest on the eye and it's got some lovely touches and features eg tells you the name of the opening you're playing and switch sides easily (what would the machine do in my shoes?). I am NOT affiliated to CG, just think it's awesome!
When I said to play against real life players I obviously meant online It never even occurred to me that you could play chess with a board against someone else face to face. It's been soooooo long since I've done that, that it didn't even cross my mind. I play on the desktop, but I think I could go for some iphone chess gaming. Though it would have to be rated and have enough players to be able to find people of similar ELO ratings (otherwise it's no fun). Is there something like that on iOS? (Preferably that doesn't require a paid subscription)
Recommend you 3D Magic Chess. It's battle chess with animations. And HD version for iPad is also released. Best of all, there are many levels fit for both beginners and top players. Also, 3 modes are available: Pass N play, PVP, Player vs AI. Everybody, give it a shot. You will like it.