The Greeks were made up of a number of city-states such as Sparta and Athens. Typically these city-states fought each other, but they united to fight against the Persians.
The Ionians were Greeks that lived along the coast of Turkey. They were conquered by the Persians. When the Ionians decided to revolt they asked Athens and other Greek cities for help. The other Greek cities sent ships and weapons, but were quickly defeated. The Persians didn't like this and decided to conquer the rest of the Greek cities in order to keep them under control.