I'm British. I only have a very vague memory of the game from my childhood. I didn't remember what the goal was, but I remembered you have to hit the opponent's one. I don't remember if I ever played it or not.
You hold the conker in one hand and the string in the other with some tension and then release so it pings and bashes the other players conker, hoping to smash it off the string.
Repeat until one conker is smashed into oblivion.
If your conker wins against multiples it becomes named mythically: a twoer, a threer, and so on.