In the time of the persian wars there were three major empires: Egypt, Greece, and Persia, who was the strongest of them all. Persia was about to have the greatest challenge of all time. First the Greeks would start the war with the Ionian revolt, then the battles of Marathon and Thermopylae, and finally they would go on the offensive for revenge.
Ionia, which was a Persian controlled city in Greece, revolted. This took place in 499-493 BCE, and started the Persian wars. As you know the Persian were, at that time, the strongest empire, and Greece was still big but not as big and strong as the Persian empire. So this must have seemed like a certain victory for Persia. In the battle of Marathon, the Athenians and the Persians clashed, it was a successful battle for the Greeks, which essentially marked the decline of Persia and the rise of the Greek empire. This also shows that even the strongest can lose.
The Battle of Thermopylae took place in 480 BC and was a very interesting battle. A much bigger Persian empire went through the mountain pass of Thermopylae, and confronted Greek forces, which were mostly the spartan forces that were led by Leonidas. After three days of holding their ground from the Persians, king Xerxes the first out flanked the spartan army. In the end the Spartans were defeated but some managed to escape before the other army had arrived to flank them.
In 478 BC the Greeks took the offensive, but Sparta pulled out of the alliance. The Delian League was formed by Athens afterwards. The states in the Delian league could contribute with men or taxes. In the end the Persians surrendered and the Greeks had one. The Persians were not to sail inside of Greek waters anymore.
This war clearly showed that the most powerful army can be destroyed by a supposedly weaker one. Also the Greeks benefited a lot from this war, yes they did have losses, but in the end they had a pretty good bargain with the Persians.