At the beginning of the 5th century B.C., the Persian Empire started a huge campaign for the conquest of Greece. They attacked the Greeks because the Greeks interfered earlier in Persian affairs in Anatolia.

A fleet under Darius I sailed from the Phoenician ports and headed west. In 492 BC, that fleet was severely damaged by a storm off Mt. Atos. In 490, the fleet disembarked at the bay of Marathon where a much smaller Greek army was waiting for them. Under the command of Miltiades, the Greek marched head on against the huge Persian infantry. Seeing this as an act of suicide, the Persians advanced with confidence. The, in a brilliant maneuver, the Greek center retreated. The Persians, seeing this as a sign of cowardice pursued, but they were caught in a pincer movement when the Greek center turned back and attacked, with the help of their flanks which were already engulfing the enemy. The Persians were defeated and fled to their ships and sailed to Athens. BUt the Athenian army marched to Athens to meet them. Athens was ready because a runner was dispatched by Miltiades to warn the city (the first "Marathon"). The Persians turned back to Persia.
The Battle of Marathon (click to enlarge)

Ten years later, Darius' son, Xerxes, sailed against the Greeks again. A large land force was also sent overland to join with the fleet. The land army conquered much of northern Greece and Macedonia. But, in 480 BC, the army of Xerxes was halted for days at the pass of Thermopylae where a handful of Spartans held valiantly with their king.
The Battle of Thermopylae (click to enlarge)

By that time, the Athenian fleet was ready and defeated the Persian fleet at Salamis. The Persians turned back and the land force remained for a time. That force was eventually destroyed at the Battle of Plataea in 479 BC.