The Dominion was a political entity native to the Gamma Quadrant. It was created and ruled by the Founders, though most of the administrating of their empire was left to the Vorta. The Dominion's overriding goal was to fulfil the Founders' will and bring order to the galaxy.
The Vorta and Jem'Hadar were genetically engineered to believe that the Founders were gods. To the other members of the Dominion the Founders are an almost mythical group due to their self imposed isolation from interacting with other species.
The Dominion covered a large area of the Gamma Quadrant and it maintained a massive military made up primarily of the Jem'Hadar, who were genetically engineered soldiers.
The Dominion became aware of the United Federation of Planets and other powers from the Alpha Quadrant after the discovery of the Bajoran wormhole by the Federation. The Dominion turned its attention to the Alpha Quadrant and spent several years destabilising the region before it backed a military coup by Gul Dukat, who in returned joined the Cardassian Union to the Dominion. This gave it a significant foothold in the Alpha Quadrant through the absorption of the Union's military and industrial might, which it also supplemented with larger convoys of Jem'Hadar and Vorta. (DS9: "By Inferno's Light")
In late-2373, the Dominion went to war against the Federation and Klingon Empire. (DS9: "Call to Arms") The inclusion of the Romulan Star Empire against the Dominion and the complete lack of military support from the Gamma Quadrant due to travel through the wormhole not been possible saw the Dominion defeated after two years of fighting. (DS9: "In the Pale Moonlight", "What You Leave Behind") With the war over the Dominion forces retreated back through the wormhole into the Gamma Quadrant.
In an alternate universe the Dominion won the Dominion War after the failure of Starfleet's Operation Return. With the supply lines to the Gamma Quadrant open the Dominion quickly overwhelmed most of the Federation and soon after the Klingon Empire bringing an end to what they would rebrand as the "Unification War".
Following the war the Dominion accepted Bajor as a member. As part of their agreement with Bajor the Dominion would not station forces in their system, though a representative would be station on Terok Nor. A year after that they handed back Terok Nor to the Bajorans as they had established their own space station on the GQ side of the wormhole. For years to come they would hold up Bajor as an example of how peaceful integrity with the Dominion could have great benefits.
In 2394, following the Bajoran Reckoning the Dominion moved into the Bajor system to investigate the disappearance of the Bajorans and loss of Gateway Station. They also discovered the Bajoran wormhole had disappeared. They planned to build a new station at the site of the wormhole and restrict access to the system to keep the loss of the wormhole a secret for as long as possible. (Star Trek: The Farthest Star: "The Final Prophecy")