Step 1: The conker
The conker got its name from the game, rather than the other way round - before the horse-chestnut was introduced to Britain, the game was played with acorns or snail-shells (the word conker actually means "hard", and comes from the same root as the French conque, meaning "conch (shell)").
Step 2: Preparation.
Being a playground game, the drilling has been done with many tools, typically screwdrivers and found nails, and the string is often a shoelace.
The neater and rounder the hole, though, the less likely the conker is to split. Use of a power drill is therefore recommended, as seen below, with #2 son helping me prepare for a Cub Pack contest.
The drill-bit used should match the string being used - thread the conker onto about 30-45cm (a foot to 18 inches) of string, then tie a good fat knot in the end of the string to stop the conker slipping off.
Shoelaces have an advantage, with a built-in threader (the aglet); you may have to resort to a large needle, awl or bent paperclip to thread the string through.