Rune Raiders isn't a new app -- it came out a little while ago and has been updated a few times (and you can actually play a version of it online in Flash). But I was looking through the App Store for a few good turn-based games recently, and Rune Raiders has been stealing more and more of my playtime. It is indeed a game you play by turns, where you control a small party of square-shaped soldiers, who move upwards on a grid-based battlefield, fighting a series of enemies who are also represented by tiles.
Each of your heroes (and the enemies you face) have various abilities and uses, and each of those abilities has their own range and strength of attack. You can push your party to the right or left or forward, or simply move them at will around the bottom of the screen. Of course, every move takes up a turn, and the game progresses in this way, with you trying to make the fewest turns to keep your heroes alive and bring them to the end of the level as quickly as possible.
The whole thing is governed by a currency system (supported by in-app purchases), and unfortunately, the system is a little wonky -- if your characters die during, you can spend some more money to get them back to life. That's a little more annoying than it has to be, I think -- I wish the devs here had come up with some other way to keep characters going than simply asking you to pay more money.
But if you're smart with your moves and the strategy behind them, it's completely possible to get lots and lots of gameplay in just by using and earning the in-game currency. There's also a rating system for each level, and you open up different heroes as you go along, so there's plenty of replay value as well. Rune Raiders is a free universal download that's available right now.