The ultimate cause of the Peloponnesian War was the fact that Athens and Sparta were the two "superpowers" of contemporary Greece. Given their rivalry, it was essentially inevitable that they would "come to blows."
The more proximate cause of the war was conflict between Athens and two of Sparta's allies, Corinth and Megara. As Corinth and Megara came into conflict with Athens, they called on Sparta for help. They even threatened to withdraw from the Peloponnesian League (headed by Sparta) if it did not help them. To protect its alliances, Sparta joined the war against Athens.