A Horse Chesnut.
Basically, you knock it out of the tree (or collect it off the ground) bore a hole in it, put a string through, and hold it out at arms length. Your opponent then tries to break your conker with his, by swinging at it. Each person takes turns, until one of the conkers breaks. The winning conker is then known as a 'one-er', after 2 hits it is a 'two-er' and so on. Obviously, a hard skin is essential to a successful campaign, so various tactics, such as soaking in vineger, and/or oven baking are sometimes used to get 15-ers and above....
Err... can you tell I'm bored in the office today?