I agree that Shredder is excellent for iphone and has a huge number of features, but Hiarcs has recently been updated and is, in my opinion, probably slightly better for a hardcore chess player. I'm more of a casual user and my 'wants' list may be shorter than a club player's, but I appreciate the stats and opening book advantages that I think Hiarcs has over Shredder.
Shredder does have the inbuilt tactics puzzles though; unlike aidin I can't find the tactics puzzles he believes are in Hiarcs, which I own on both iphone and ipad - maybe I haven't searched hard enough yet.
The other chess app that is a must on iphone and ipad is T Chess Pro.
My ipad 2 likes this as its the only chess engine on the platform that allows dual core processor utilisation. Not that I need really it.....but I guess its nice to know its being used and to be honest the speed improvements arent something I notice - the engine slaughters me just as quickly with the option turned off, but oh well....
I wouldn't bother with Fritz at all - it's slow and graphically a divsion below the aforementioned three apps. If you have an ipad as well as an iphone, T Chess Pro is the best value, as its a universal app and so only needs purchasing once. Check out some reviews and make up your mind from those, but I'm happy owning all three and having the choice to switch between them all.
I contemplated buying Deep Green Chess, which looks rather nice but ultimately doesn't offer anything like the number of features of the others; if it were cheaper then fair enough, but all of these apps are around the £5 or £6 mark and so most people wouldn't want to spend that amount on all of them when one would likely suffice.
One other engine/app worth considering is Chess Wise Pro. This is decent value, as its universal and also offers a very classy looking FICS interface into the bargain which looks lovely on an ipad. Aesthetically its a real treat, understated and just simply well designed.
As for Tactic Trainer (namechecked here, too -
http://www.guardian.co.uk/sport/2011/jul/11/chess-ipad-apps-tactics ) just type Tactic Trainer into the itunes app store. I think it claims to have over 20,000 problems. Again, a universal app.
Hope some of this helps.