38i has created a cool game here which is a Korean classic called Kono.  Your goal is to jump your peg over another one of your own colored pegs and land on the opposite color.  It's pretty simple once you get going but the computer's smart on Advanced so play smart and think ahead.  You can also set the difficulty level to Beginner or Expert.

I definitely liked Kono and will be playing it again.  The idea is simple but the game is complex. 

Your objective is to turn all the lights off but, it's not that simple because turning one off can turn another one on! Lights off is a mindbender and really fun.

The PowApps instructions say:

Just Tap on one light and you will switch it on or off, along with the four adjacent lights. You win after getting all of the lights to shut off.

Arrange the balls of the same color in straight lines of 5 or more. You can move the balls around and longer lines get more points. It's a little tricky at first but this is a great game.  It runs great and is definitely one of the better games made for the iPhone.  The info and instructions say:

To move a ball, click on it with the mouse (circle will appear) to select it, then click on a free cell in a field. To deselect a ball just click on another. If movement is possible, ball will go to the specified empty field. After each turn, 3 new balls are placed on a field (random colours & positions). Preview panel shows colours of the next 3 balls. You can alway force the next turn without moving anything. Click on preview panel. If the game field is full, game is over. Refresh page to start a new game.

Game goal is to arrange balls of the same colour in a straight lines (every direction). 5 or more balls on a straight line are removed and points are given. Longer lines give more points to you. Crossing lines count together.

You do not have to stay online during the game. You can disconnect from internet after complete page loading and play entirely offline.

Logic3 requires you to think, and think hard. Your objective is to get all the buttons to say 0.

There are a bunch of difficulty levels so I'd recommend starting out on 'Real Easy' or something like that. 

Lumina Memory Game is an extremely high quality Simon style game for your iPhone.  CrossComm has created this game with audio and that really helps to keep track when you get more advanced.

It runs smooth and looks great.  Definitely play this one.

This is a very clean version of sudoku. It's well designed for the iPhone that has four difficulty levels and allows you to undo any mistakes.  The site's instructions say:

Select a puzzle difficulty and then play. To fill in a cell just touch it. You'll know if you have made a mistake.
If you slip, you can touch the cell again to blank it.

Create the evolution of man by moving the panels around into the appropriate order.  This is an interesting game that's fairly fun.  The MacPips directions say:

The evolution game is very simple to play but hard to win!
Your objective is to show the evolution of man in the four windows, or if you prefer, get four images of Michelangelo's ‘Creation of Adam’ in a line.

This is a smart little game of moving colored pieces from peg to peg.  You can't have more than 7 on one peg at a time and the colors have to match up. Be sure and read the instructions at the bottom or you might get confused.

The idea is simple. You must get all of the pieces onto their respective coloured pegs, but the colour of the piece you are moving must match the piece you are moving it to. If moving to an empty peg, the piece must match the colour of the peg.

You can move one or two pieces at a time and you can't have more than 7 pieces on a peg. Click on a piece, then on the place you want it to go to move it.

38i has created a slick, fun version of Mahjong which plays well and looks great.  I've always been a fan of this classic and there are a few versions available so play this one and then try iMahjong.

Either version is great though so if you've played Mahjong in the past and enjoyed it you'll love this. 

Click the green guy to move it around the maze. Get out and you win!

Instructions: Click on the screen to move the green circle around the maze. Try to get to the exit at the bottom right.
Each time you play, a new random maze is generated.