When the heavens and the earth were created, and man and woman were placed on the earth, the creative order was established. (Please click here.) This illustration depicts that the "dominion" was the created order through which the creator exercised his will over creation. Of course this order started with the creator, and then to man/woman (one flesh), and then to the animal kingdom. You can call this created order God's "chain of command."
When the serpent approached the woman and deceived her, and then the man had subsequently sinned, this creative order of God was turned upside-down. (Please click here.) Please note that the serpent was Satan (according to Revelation 12:9 and Revelation 20:2). By approaching the woman as an animal (instead of appearing as himself), he flipped the creative order upside-down.
The same created order (in the positive sense) was repeated with the nation of Israel. In the Hebrew Bible, the visible Kingdom of God on earth was represented by the theocracy, which was headed by the Shekinah Glory, which was the physical presence of God on earth. The King of Israel was the head of the nation of Israel, and therefore was part of its body. In this arrangement, the Gentile nations were the "animals." (Please click here.) As in the case of the created order in the Garden of Eden, disobedience came along and again flip-flopped the created order. When the Shekinah glory left the earth before the Babylonian exile, the prophet Daniel described the rulership of the world through the domination of "beasts" or Gentile kingdoms. Since the departure of the Shekinah glory from earth, the world has been governed by "bestial" kingdoms, and so began the Times of the Gentiles (cf. Luke 21:24). Is it not remarkable that nations of the world today depict themselves through the image of predatory wild animals (such as the bald eagle, the bear, or the lion)?
Finally, and not least, we are now living today in what is the invisible Kingdom of God on earth. During the present time (which is the Times of the Gentiles), the rule of God on earth is invisible, and is operative through Jesus Christ and his church, which are one body (Romans 12:5). At a time in the yet future, the church will become apostate (2 Thessalonians 2:3), at which time the flip-flop will occur again. (Please click here.) That is, the faithful remnant (minority) will be removed from the earth, and the apostate remnant (majority) will remain, and will "ride on top" of the beast (Revelation 17:3), who is the mega-beast empire, whose head is the Antichrist. The apostate church will be "married" to the Antichrist and be one body with him. By desecrating the yet-future temple, the flip-flop is consummated: the blasphemy is the subordination of the Father in heaven in a reversal of God's highest created order on earth (Christ and the Church), which will then be headed by Satan.
So, to answer your question, "dominion" means the creative order (or God's "chain of command"), through which he exercises his holy will within his creation on earth. Of course this "dominion" has its opposite in sin, when disobedience is exercised contrary to the will of God. The sinful version of the created order turns everything upside-down, and of course has its ultimate origin in the serpent ("beast"), who is Satan.