The war began because of some major differences between the two biggest Greek city states at that time: Sparta and Athens.
OK., so now we know what the differences between Sparta and Athens were - but that doesn't tell us why the war began in the first place. First of all, there were many reasons, but the main one was the argument between Athens and Megara. Athens prohibited trading to Megara, because they were their competition. Athens were a part of Deliand league and Megara was in Peloponnesian league (an ally of Sparta). The people of Megara wouldn't survive without trading since it represented their primary income, so they went to Spartans. Spartans then demanded the cancellation of prohibition from Athens, which refused to cancel it. Sparta then declared war on Athens (431 BC).
In the year of 430 BC, plague (disease) attacked Athens and killed one third of their population. The main cause for that to happen was, that they were all hiding behind the Long wall of Athens and due to high capacity of people in one place, it was easy for the disease to spread from person to person.
In 421 peace was made between the city states: Peace of Nicias. It was set for the next 50 years.
But that didn't end the war - Athens broke the peace in 415 BC when they attacked Sicily, who was an ally of Sparta. The main city state there, Syracuse, defended very well until in 413 BC Spartans came and defeated Athenian army and fleet. The war continues until 405 BC, when Athens are beaten and forced to capitulate (surrender). They then agreed to:
- only have 12 ships
- put down the Long wall of Athens
- disband Delian league
- accept oligarchy (only the rich were in power)
Now Sparta ruled all the Greece, until 362 BC.