(covers information from several alternate timelines)
A consort was the spouse of a ruler. More generally, the term referred to anyone who shared the lot of another, but especially a partner or spouse. Historically, the consorts of rulers generally had no power unless it was given to her by her husband or by a law.
In 2269, criminally insane ex-Fleet captain Garth of Izar chose Marta, a dangerously insane Orion woman, as his consort as part of his self-styled "coronation" as "ruler of the galaxy" at the Elba II asylum they and their fellow inmates had taken over. Shortly after, however, Garth punished Marta for kissing Captain Kirk by having her brought outside the facility, exposing her to the toxic atmosphere of the planet, then detonating a small sample of a powerful explosive he had developed and inserted into her necklace. (TOS: "Whom Gods Destroy")
In 2366, a Tasha Yar from an alternate timeline returned with the USS Enterprise-C to the battle at Narendra III. The ship was destroyed by the Romulans, and its surviving crew captured. A Romulan general who oversaw their capture became enamored of her and spared the lives of the remaining crew in exchange for Yar becoming his consort. One year later, she gave birth to a daughter named Sela. (TNG: "Yesterday's Enterprise", "Redemption II")
In 2371, Seska, an undercover Cardassian operative, was stranded in the Delta Quadrant on board the USS Voyager. In order to get home, she attempted the transfer of Starfleet replicator technology to the Kazon-Nistrim in exchange for their protection but was caught. She escaped aboard a Kazon raider and became the consort of First Maje Culluh, the leader of the Kazon-Nistrim sect. One year later, she bore him a son. (VOY: "State of Flux", "Basics, Part I")