Actually, Athens was part of Greece as was Sparta. The Peleponnesian wars broke out following the final defeat of Persia during the Second Persian War. The absence of Persia had created a power vacuum in the Aegean basin which both city-states were anxious to fill. Athens created the Delian League(named for the sacred island of Delos) and required member states to contribute to the Athenian navy. City states were not given the option of not joining the league; and Athens ruthlessly enforced its control, often waging war against its own members. Sparta, alarmed at the spread of the apparent Athenian empire, formed its own league known as the Peleponnesian League. An uneasy truce existed between he two until the city of Syracuse attempted to abandon the Delian League and join the Pelopnnesian league. When Athens attempted to force Syracuse militarily to remain within its league, Sparta responded and the war broke out.
Among the instances that led to the outbreak of the war was the defection of Alcibiades, an Athenian leader and relative of Pericles, to Sparta. Shortly after his disappearance, a number of statues of Hermes within Athens were emasculated, a grave and sacrilegious insult. Sparta and Alcibiades were blamed for the vandalism, and this was the spark that ignited the war.
For further information, you might consult Donald Kagans Peleponnesian Wars.
The Peloponnesian War lasted for almost thirty years and concerned the Spartan invasion of and eventual victory over Greece. The name comes from the constant attacks on the Peloponnesian peninsula (home of Sparta) by Greece, and that location's central role in launching and repelling attacks from both sides.
As the city of Athens gained power and influence over its relative city-states, Sparta became the only one to repel its attacks. After Greece defeated the Persians, who had tried to invade, they launched an invasion of their own. During this time, Athens increased its power and territory so much that it became an empire, ruling over almost all the other Grecian territories. Other states began to feel oppressed, and the last truly free state, Sparta, encouraged rebellion inside Greek borders. Battles among the various states, always overshadowed by tension between Athens and Sparta, kept the region in a state of war until the final declaration by the Spartan assembly of grievances against Athens, officially starting the Peloponnesian War.