Governing an empire is probably tough work, how did these people do it? Did they assign regional governors? Were they often selected from the governed people or were they assigned by the governing culture? How did these empires enforce compliance and regulation throughout these areas?
It is notoriously difficult to say anything with certainty about the internal workings of Achaemenid Persia. Sources are extremely scarce; most of them either hark back (partly) to earlier Assyrian templates, which makes them hard to interpret, or come from Greek sources, which usually implies a strong bias. Only a few Persian inscriptions and no Persian works of literature survive. For the Medes, the situation is even worse. We know so little for certain about them that there is a running scholarly debate over whether they existed at all.
That said, there are a couple of things which are attested in literature or archaeology about the way Achaemenid Persia worked. Mostly, it seems to have been a network of administrative regions known as satrapies, which each had a satrap appointed by the Great King. These were often nobles from the royal family or from other Persian families with close ties to the ruling house. Each satrap had primary responsibility over the defence and public order of his satrapy, and was required to deliver a certain sum in taxes and tribute to the Great King. Should he fail, the King could either send a replacement (which often meant that the replaced satrap was executed) or gather his army to set things right in person.
Satraps were supported by a network of local lords, and had garrisons stationed in key cities and forts. Whenever the need arose, each satrap was required to levy troops from his satrapy and send them to a previously determined rallying point to make up the royal army. This is probably the reason why Greek sources refer to some officials as "satrap over all who muster at [X]" - their domain was determined by the catchment area of their local mustering place, from which they would set out to join the King's host.
Parts of the empire were not satrapies but semi-autonomous regions ruled by vassal lords, while the city of Babylon was exceptionally allowed to retain its own administrative system.
Although we haven't been able to find it, most scholars believe Herodotos' claim that the Persians built a Royal Road from Sardis to Susa, and possibly east beyond that, with waystations and fortresses along its length. This served to speed up royal communication and mobilisation across the vast empire.
I'm told it's partly outdated now, but the main textbook on Achaemenid Persia is still Pierre Briant's From Cyrus to Alexander: A History of the Persian Empire. Worth a read if you'd like to know more.