Were the Athens& Spartan like 2 political parties within the same country, or 2 states in the same country?

Hi, I know that the Sparta and Athens were 2 different Polisis (city states) but the way my teacher explained the Greek/ Persian war made it seem like Athen and Sparta were a team? could someone clarify this for me? was the war with the Persian empire after the Sparta/Athen war? Thank you!