They are parasites, like lice, or something. When I was a kid, according to most girls, most boys had "Cooties".
EDIT: They were lice.
"The original cooties were very real and extremely nasty, since the word was first applied to body lice. It's a slang term intimately (and I mean that sincerely) associated with the military in World War One. It's first recorded in print in 1917, but is presumably older."
In the game, whoever got his Cootie put together first, wins. You had to roll dice, or something, to be allowed to attach one piece (leg, eye, etc.) to the body.