In English translation, several different Greek words are translated as "kingdom" but they are all based on the same root.  All relating to a reign or realm of a ruler. The same form of the word can be the singular of the word meaning "realm" or "reign" as a subject or the plural of the word meaning "palaces." as a subject or object.  "Palaces of the skies" is an especially appealing phrase, but the plural form doesn't often fit with verb used. 

The Greek Words

The kingdom" is from the Greek word basileia, which means "kingdom", "dominion", "hereditary monarchy", "kingly office," (passive) "being ruled by a king," and "reign." The word translated as "kingdom" can be the region, the reign, the castle or the authority of a ruler. Christ does not seem to use it to mean a physical region, so its translation as "reign" or "realm" seems more appropriate. This is especially true because the "reign" of a king means the execution of his will.

Speaking linguistically, the Greek word translated as "kingdom" means that which is controlled by a central authority. It is generally the rule of kings, but it was also applied to the Archon of Athens, who was elected. It can mean the place that is ruled, the people who are ruled, or the person who is ruling (if she is female because the word's form most common in the Gospels is feminine). It can mean the capital city of an empire or the ruler's castle. Our English word "basilica," meaning the seat of power for a bishop comes from this word. Generally, it refers to the concept of hereditary rule, the passing of authority from one generation to the next.

The Greek "kingdom" is more specifically under the authority on one man or one house, (see this article on "house") in English, the term can mean a nation and a people. The word, like our English one, comes from the Greek word translated as "king," basileus, which means a "king", "chief", "prince", "lord", "master", "a great man," and "the first and most distinguished of any class." This is the top person in a region of any size. 

From Jesus's viewpoint, a kingdom has one leader and must be united behind him. Beneath the kingdom, a nation is divided into "houses" where allegiance goes to the master of the house. However, the king has authority over all the houses below him.

Jesus Usage

Though it is consistently translated in the NT as "kingdom," we can think of it as a "dominion", "that which is under a central authority", "being controlled by a leader," or, more simply as "rule" or "reign." In English, the word that comes closest to this idea is that of a "kingship" since it includes both the realm, the authority, and the reign of a king. 

Of course, the most common reference Jesus makes to a "kingdom" is the "kingdom of heaven" or, more accurately, "the realm of the skies." See this article about that phrase.  He also refers to the "kingdom of God" or more accurately, "the realm of the Divine."

However, the word translated as "king" is broader than our "king". The word always means that which is controlled by a specific person, a "basileus," which means "leader", "prince", "commander," or "king." Basileia is not a synonym for a state, a country, or any social group of people though we often think of a "kingdom" in those terms. A basileia is defined by its control or ownership by the master or leader and refers both to people and property under that control.

For example, the Catholic Church who follows the pope in Rome is a basileia. This "kingdom" includes not only the Papal States but all the property owned by the Catholic Church worldwide. It also includes all the people employed by the Catholic Church. As we extended it to mean the members of the Catholic Church, however, the definition gets into a gray area. Church members are not really controlled by the pope but they do follow the pope.

Would Christ have recognized other of our modern organizations as a basileia? Do the United States, Apple Computer, or the United Auto Workers qualify as "kingdoms." They all have a central executive leader that is the controls the organization's people and property. However, in these kingdoms central authority more limited in terms of controlling their members because these leaders' word is not "law" except within boundaries. Again, this lack of complete authority gets us into a fuzzy area.

However, since the question here is about the "kingdom of heaven," these gray areas are all swept away. The ruler here is clearly the Divinity, God. When Christ says "the kingdom of God," the meaning is clear, but when he says "the kingdom of skies," we can also be pretty certain that the king he is referring to is God. We are pretty certain that the ruler is not "the skies."