As a verb, recount can mean either "tell the story of" or "add up again." As a noun, recount usually refers to the second (or third or fourth) tallying of votes in a close election.
The word count comes from the old French conter, which means "add up" or "tell a story." Here's a story to recount (narrate): In an election featuring Count Dracula and Count Johnson, Dracula wins by two votes, so Johnson demands a recount (an adding up of the votes again). The officials recount (count again) the votes, and this time find Count Johnson the winner. Dracula threatens to bite the officials and they quickly change their minds.