This is a simple checkers game built out of relatively common elements. I tried to use as few pieces as possible. You will need:
~ 1 16x16 baseplate, or you can build one out of regular plates
~ 32 2x2 plates (if you want to do two colors, then you'll need 32 of each color)
~ 16 round 2x2 bricks (8 of each color)
~ 1 technic crossbeam (it should be reasonably long, like the one I use)

Step 1: Place the Tiles

Arrange the tiles in a crisscross pattern, leaving holes where the base plate shows through. If you want to use two colors, then place the second color over the holes in the pattern.

Step 2: Place the Game Pieces

arrange the pieces in the same way you would in a normal checkers game. You're done building!

Step 3: Playing the Game

The rules work like a normal checkers game, just with the size cut in half. To king a piece, just place it on top of a captured piece.

Step 4: Getting the Pieces to Move

As you play, you'll find out quickly that the pieces can get pretty fond of their tile, and don't want to leave it. This is where the cross beam comes in. There are two techniques to remove the piece easily:
1) Use the crossbeam and the hole in the piece to move it. Place the crossbeam in the piece, then push down on the beam while pulling up on the piece. It pops right off.
2) Press the crossbeam or your finger on the corner of the tile as you pull up on the piece.