Zinc and Copper, and all metals in fact, conduct electricity through their electrons. Metals are elements that have a lot of "loose" electrons on the outsides of their atoms. These electrons can move if connect a battery to the metal. This is how electricity is generated - A battery is hooked up to a wire (made of a metal, usually Copper), and the "loose" electrons get pulled from one side of the battery to the other.

Based on your next question, I'm guessing you're asking about how Zinc and Copper can be used in a lemon battery. In this setup, the Zinc is there to "donate" electrons to the Copper. These electrons flow through the wire connecting them, and we get