There are varying rules for a conker fight, but these are the ones laid out by The Woodland Trust, so you can run your own back garden championships:

Wrap the loose end of the string around your hand. One player should dangle their conker on about 25cm of string, keeping it absolutely still.
The other player then swings their conker at it to try to break it.
If the attacking player misses, they can have two more chances before it’s their opponent’s turn.
Take turns until one of the conkers breaks and you have a winner.
A new conker is called a ‘none-er’ as it hasn’t beaten anyone yet. When it beats another conker, it’s a ‘one-er’. If it beats another, it becomes a ‘two-er’ and so on.

Turn to page 64 of October’s The Simple Things for How to do autumn well, including how to pick a prime pumpkin, Apple Day, and how to go mushrooming (without updating your will).