It was between an alliance of Greek city-states and the Persian Empire. Today we call it the Persian War to differentiate it from the wars between the Greek city-states. It is often called the Greco-Persian War.
satrapies
The Persian thought the Persian Empire. The coalitions of Greek city-states thought the coalitions of Greek city-states.
Countries with a coastline on the Persian Gulf, called the Persian Gulf States or the Gulf States, are (clockwise, from the north): Iran (Persia), United Arab Emirates, Saudi Arabia, Qatar on a peninsula off the Saudi coast, Bahrain on an island, and Kuwait and Iraq in the northwest.
It was initiated by the Persians against the Greek city-states around the Eastern Mediterranean, so Persian is a convenient name to give it. Some call it the Greco-Persian War - you will find a summary of it online in Wikipedia under that title.
Persian Empire.
The Peacock throne is called "Takht-i Tāvūs" in Persian. The peacock throne was called Takht-i-murassa in Persian.
They called it the Kingdom of Persia. We call it the Persian Empire.
Which Persian War? There were ancient Persian Wars (Persia doesn't exist anymore by the way), Iran used to be called Persia. Which Persian War concerns your question.
Oregano in Persian is called "مرزه".
The Persian army, and the armies of the Greek city-states.
The Persian army, and the armies of the Greek city-states.