The Dominion consists of a number of subordinate races in the Gamma Quadrant - claimed to be "hundreds" of species on "thousands" of worlds by the Dominion leadership, although only a handful of these were seen. It is ruled by a race of shape-shifters who refer to themselves as the Founders or Changlings. The Founders claim to have suffered severe discrimination at the hands of "solids" in their past, so their governmental policy is to subjugate all solids they encounter in order to protect their race. A large-but-unspecified number of Founders join to form the Great Link, which is effectively the ruling body for the Dominion.
The Founders set policy for the Dominion and conduct covert operations in foreign territory, but they handle little of the day-to-day operations of the Dominion. They have genetically altered some races from the Gamma Quadrant to serve them, including the Vorta -- who serve primarily as administrators -- and the Jem'Hadar -- who provide most of the Dominion's military personnel.
The Founders are the unquestioned rulers of the Dominion. Their genetically-engineered servant races treat them as gods, and the power of their armies and fleets cows all other opposition in their territory.
The Vorta are the administrators of the Dominion. They are genetically engineered to worship and serve the Founders. No Vorta has ever willingly betrayed the Founders, although one did disobey orders because he believed his actions were in the best interests of the Founders.
The Jem'Hadar comprise the military of the Dominion. Although genetically engineered like the Vorta, Jem'Hadar loyalty is not as reliable, so all Jem'Hadar are addicted to a drug called Ketracel-white, which only the Dominion can provide.
The Dominion controls a large amount of territory in the Gamma Quadrant, which is "thousands" of worlds according to the Founders. The best estimate is between 5,000-10,000 conquered worlds with only a few hundred member worlds. Their territory may exceed that of all the major Alpha Quadrant powers combined. Despite this, they are pitfully small compared to the Empire.
The Dominion are a major power in the Milky Way galaxy, but their influence is restricted to the Gamma Quadrant. Before the closure of the Bajoran Wormhole, they presented a dire threat to the sovereignty of all of the major powers in the Alpha Quadrant.