How did Persian kings maintain control and organize such a vast empire?
Basically, the Persian kings were able to organize and control their empires because they were willing to delegate authority and to pay some attention to the desires of the people they conquered.
The Persian empire was divided into "satrapies," each ruled by someone called a "satrap." Satraps were essentially kings in their own areas, but the Great Kings tolerated this, which allowed the system to work, thus making the empire easier to administer.
At the same time, the kings kept their subject people happy by letting them keep their own religions and even funding temples for religions of the conquered people. Similarly, the kings allowed many of the elites of the conquered areas to hold positions of power. By doing things like this, the Persians avoided acting simply as conquerors and made their rule more popular. This helped them to control their empire.
The Persian kings governed their massive territory by dividing conquered lands into provinces known as "satrapies." Each satrapy had its own governor known as a "satrap" who was appointed by the king. These officials were responsible for collecting taxes, maintaining a military, and had supreme judicial authority over their region. Satraps reported directly to the king and were periodically inspected by Persian royal officials. The royal officials were referred to as "trusted men" who assessed taxes and recruited able-bodied males for the army. Two other positions that were essential to maintaining an efficient government were the "Royal Treasurer" and "Garrison Commander." Satraps worked with both officials who reported directly to the king. The central administration was also responsible for building and maintaining new roads. This expansive system of roads allowed merchants to travel freely which enhanced and encouraged trade throughout the empire. In addition to the various trade routes, the Persian monetary system introduced by Darius created a more organized economy.
The Persians often get a bad reputation in the West, but if we examine their vast empire, we cannot but be impressed by what they were able to achieve. They had three main ways of ruling.
First, like the above answer intimated there was a system of local governmental officials, called satraps. These satraps ruled over a region called satrapies. This hierarchy was effective in organizing vast areas. Like the Romans, the Persians were organized and structured.
Second, it must be stated that the Persians had a vast army and navy. We must not allow the Greek defeats of the Persians cloud our vision of the military power of the Persians. The Greeks did not defeat the entire Persian army. This point is important to mention, because people for the most part feared going against such a powerful army. This alone helped the Persians to rule.
Third, the Persians adopted a policy of freedom. For example, they were very sensitive to the people they overcame in important matters such as religion. Even a reading of the Old Testament/Hebrew Bible shows that the second temple was built under the encouragement of the Persians. For more information read, the books of Nehemiah and Ezra.