A conkerer plays the British game of conkers. A brief description may be desirable for those unfamiliar with it. Conkers has two players, each armed with a nut of the horse chestnut threaded on a string. Players take turns hitting their opponent’s nut with their own. The player whose nut breaks first loses.
Conker is a dialect word that originally meant a snail shell, with which the game was first played, though without the strings (it would be classed as animal cruelty these days, as the shells were frequently still occupied). Frederick Ross described it in A Glossary of Words Used in Holderness in the East-Riding of Yorkshire of 1877: “In the boy’s game of conkers the apexes of two shells are pressed together until one is broken, the owner of the other being the victor.”
Strangely, for a game often considered to be an immemorial part of the English annual round of boy’s games, the snail sense is first recorded only from the early nineteenth century, and the horse chestnuts one from the 1880s.
The word might be from conch, but could equally well be a respelling of conqueror, since the game was often spoken of and spelled that way in the nineteenth century.