It's connect 4 again! This is a slick version which will work without an internet connection once you play it once. I absolutely loved this version of the classic Connect 4!  The site's instructions say:

InARow is the iPhone version of the popular board game Connect Four (also known as Four-in-a-Row and Four-in-a-Line). This has been designed specifically for the iPhone and the iPod Touch. Play against another human or play against the computer with beginner, intermediate and advanced levels. Is it possible to beat the computer in advanced mode?

I got beat by a computer. :(

No, really, this is a great version of chess which works well on the iPhone. I had a ton of fun playing round after round vs. Shredder. It's not difficult to get the hang of this, just click the player you want to move and then touch the grayed out box you wish to move them to. But be careful because Shredder's pretty good!

Reversi aka Othello is a great classic game that's now being made for the iPhone.  The wikipedia page says:

Reversi and Othello are names for an abstract strategy board game which involves play by two parties on an eight-by-eight square grid with pieces that have two distinct sides. Pieces typically appear coin-like, but with a light and a dark face, each side representing one player. The object of the game is to make your pieces constitute a majority of the pieces on the board at the end of the game, by turning over as many of your opponent's pieces as possible.

This specific version looks great and plays smooth.  If you like Othello you're going to love this. 

40,000 puzzles on 5 difficulty levels! This is your new Sudoku game and it even offers hints to help you.

The only catch to iPhonus Sudoku and iPhonus Sudoku Tournament is you need to register to play.  But, if you want a spot where you can come play day after day and keep track of your progress then definitely use iPhonus. 

This is essentially iPhonus Sudoku - tournament style.  The gameplay is smooth and high quality.  You'll need to register but with iPhonus you can compete against many people and see how good you really are.

 Have fun!

Build pipes as long as you can as fast as you can. Don't let the slime catch up to you or you'll lose! 

The game's instructions say:

The Aim
The aim of iPipes is to build a pipe a certain length, before the slime reaches the end of the pipe. Each level the pipe must be longer and the slime comes faster!
How To Play
Along the bottom of the grid is the upcoming pieces, and the piece you can currently place (highlighted). To place a piece in a square, simply touch the square you wish to place the piece in. An undo button at the bottom right allows you to fix mistakes, and the fast forward button makes the slime start flowing early.
 

Play 6 different levels of Sokoban. If you didn't know, Sokoban is a game where you push boxes around mazes one at a time.

You're the little character and you want to move the colored boxes into the empty containers in the fewest moves possible.  You can move the little guy by tapping around him.  Be sure and do the least moves though!

This game is similar to tetris but is with cute little animals. Get the animals together in a line and the disappear!  The site instructions say:

1) Puzzle speed = 80+700/level miliseconds, the smaller value the faster puzzle falls
2) If puzzles created in current level >= 10+level*2 then increase level
3) After puzzle falling score is increased by 1000*level*linesRemoved
4) Each "down" action increases score by 5+level (pressing space counts as multiple down actions)

Kingdom Game is a wonderful real life strategy game in which you build, fight, explore and spy on various kingdoms.  To play you will need to register, choose your world and name your own kingdom.  Once you're setup and going start building and make your plot as big as possible!  The information from W5 says:

In Kingdom Game you build your small estate into a thriving Kingdom with shrewd tactics, strategy and diplomacy.

Kingdom Game is a browser based game, pitting you against hundreds of other players, online in a real time ongoing strategy game.

You can build new buildings; attack and spy on other Kingdoms; build diplomatic relations through the embassy; buy and sell goods at the market.. - all while you work towards the ultimate goal of being the Ruler of the known world. 

My only wish to make this game better would be for some little icons and images for what you're building.  But it's a great game overall and you should definitely play it. 

The object is to clear the board by landing on every object. Move around by clicking on things in a row or column as your knight..who’s wearing red. Clear the objects and move to the next level.

It’s tricky but you will have fun playing this.

MacMost instructions:

The object of the game is to clear the board by landing on every object on the board. To move, click on an object in the same row or column as your knight, who is wearing red. Clear all of the objects to move on to the next level. You can change directions when you land on an object, but not the direction you just came from. Once you have landed on an object it is cleared, and is no longer there for you to land on. If you move in a direction with no objects to land on, you will move off the board and land in the moat. There are many ways to solve one board, but to get the maximum number of points you must solve the puzzle in the correct order. Begin by picking up the key which is worth 100 points, before trying to open the treasure chest which is worth 200 points. Grab the sword, worth 100 points, before trying to kill the dragon which is worth 200 points or the goblins, which are also worth 200 points. Kill the dragon before rescuing the damsel which is worth 200 points.