A nay is a "no" or "negative" vote. If you vote nay during your club's vote, you are voting against the proposal.
If your teacher holds a vote on whether to have a pizza party instead of a geometry quiz, it's unlikely anyone will vote nay. Long ago, the word nay was simply used as a substitute for "no," but it sounds odd today: "Nay, I don't need any ketchup with my fries." You can use it, however, to mean "this word is more appropriate," as when you say, "I'm upset, nay furious, that you forgot my birthday."