Puzzles
Any puzzle that requires thinking ahead and strategic manipulation can build logic skills. Mazes, sliding-piece puzzles, jigsaw puzzles, tangrams, Rubik's cubes, Sudoku and magic-square number puzzles all demand applied logic for their solution. Look at sites listed in the resources to find online versions of sudoku and other puzzles.
Board Games
Checkers and chess have always been games that reward players who anticipate all possible moves with cool logic. Similarly, games such as Clue and Battleship allow players to thrive with deductive reasoning. Some of these games can also be found in online versions and as apps for the iPhone.
Computer Games
The advent of the personal computer and the Internet has brought with it a slew of new logic-based games. In addition to Tetris, and Silver Games' challenge that dares you to encircle a kitten one move at a time, there are websites that are entirely devoted to interactive logic and strategy games. Look at sites such as Superkids, Math Playground, and Schooltime Games. Atlas provides practice logic games to help in studying for the LSAT.