It supposedly originated in Africa, and is played with pits and stones. (The boards you can buy in stores have depressions in them to hold the stones.)

There are two players. Each player has a row six pits in front of them, and one large pit, or Mancala, to the right of those pits. Thus a game board looks like this:

(_)
o o
o o
o o
o o
o o
o o
(_)

...with each o being a small pit, and the two (_)'s being Mancalas. In this diagram, player 1 is sitting at the left and player 2 at the right. The Mancala on the top belongs to player 1 and the Mancala on the bottom belongs to player 2.
The object of the game is to get as many stones as possible in your Mancala. All stones are equal in value, and once a stone is in a Mancala, it stays there.

To begin the game, four stones are placed in each pit except the Mancalas, for a total of 48 stones. Players take turns moving stones by picking up *all* the stones in any one of their pits and dropping them one at a time into each pit (*including* the Mancalas and the opponent's pits) counterclockwise. In other words, if there were three stones in Player 1's last pit (directly to the left of the Mancala), and player 1 chose to play this pit, she would drop one stone in her Mancala, one in Player 2's first pit, and one in Player 2's second pit.

There are two additional rules that make the game interesting:

1. If the last stone that you drop lands in your Mancala, you get to take another turn.

2. Capturing: If the last stone that you drop lands in an empty pit on your side, then you may place that stone AND all of the stones in the pit directly across from it into your Mancala.

The game ends when all the stones are in Mancalas. The player with the most stones wins.

Strategies:

It's a bad strategy to collect a lot of stones in one of your pits for two reasons: first, you're guaranteed to have to drop some in your opponent's pit, and second, your opponent has the opportunity to capture quite a lot of stones.

There is a significant advantage to going first, for which reason I think there should be a handicap, although I haven't determined what a reasonable handicap would be. (Ideally it would be an odd number to prevent ties.)

The fun of the game is figuring out other strategies.

I hope I was clear in explaining the game. If you want more detail, I can give it.