(coexisting on another dimensional plane)
Before serving on the Enterprise, Sato had been a teacher in Brazil. Serving mainly as Captain Forrest's woman, Sato had previously been involved with Jonathan Archer but left him for Forrest when he was promoted to a position of higher power. When Archer briefly took command of the Enterprise in 2155, she gravitated towards him again as he was then the more powerful. While making advances towards Archer, she made an effort to stab him with a dagger, but her attempt was thwarted. Later, after she and Archer had made love, she casually proposed that T'Pol, not Tucker, might have had a hand in sabotage aboard the ship, making Archer realize that he should have suspected T'Pol as well. However, T'Pol had already led an insurgency to restore Forrest to command.
Sato was aboard the Enterprise when it was trapped in a Tholian web, though she was able to evacuate on an escape pod before the ship was destroyed and was picked up by the commandeered USS Defiant. (ENT: "In a Mirror, Darkly")
Sato assisted Jonathan Archer in using the Defiant to gain power. However, before the Defiant could make it to Earth, she seduced Archer's personal guard Travis Mayweather into conspiring to poison Archer. Sato subsequently took command of the Defiant, proclaimed herself "Empress Sato", and ordered Fleet Admiral Gardner to surrender and stand by for instructions, or she would open fire on Earth. (ENT: "In a Mirror, Darkly, Part II")
"Something about... your 'maternal ancestor'."
- - Lieutenant Sato, translating a Tholian insult hurled at "Captain" Archer (ENT: "In a Mirror, Darkly")
"These people had some strange ideas about uniforms."
"I'm surprised you're not exhausted from all the beds you've jumped into recently."
"Commander Tucker told me I should give you a few pointers in that area."
"This is the Starship Defiant. If you don't surrender immediately, we'll begin targeting your cities. Please respond."
- - Hoshi Sato to fleet admiral Gardner from the Defiant's bridge (ENT: "In a Mirror, Darkly, Part II")
The mirror Sato and Mayweather winding up on top was a plot twist devised by Manny Coto (ENT Season 4 DVD audio commentary). Although Sato's success has not yet been made explicit in canon, "Vaulting Ambition" writer Jordon Nardino theorized on twitter that Empress Sato added the title Iaponius (Latin for "Japanese") to her name, and suggested that since the name was still part of Emperor Georgiou's regnal name a century later, Sato must have been successful. He further thought it unlikely that Georgiou was a biological descendant of Sato (as Sato was Japanese but Georgiou was Chinese-Malaysian), but suggested she might be a cousin, or more likely that like Roman emperors, imperial succession was determined at least in part through adoption.    
The story of Empress Sato's reign is told in the story "Age of the Empress". The ENT-era story is based on a story outline written by former Enterprise producer and "In a Mirror, Darkly" writer Mike Sussman.
In The Sorrows of Empire, the Sato dynasty continued for more than a century through the use of binary clones of the original Empress Sato, so that the rulers of the Terran Empire would forever have her face; the third Empress Sato commented that she and her "sister", Empress Sato II, were "nothing more than copies of the biggest narcissist in galactic history". The dynasty would end with the death of Empress Sato III and her entire imperial guard (courtesy of the Tantalus field) in 2277, paving the way for Spock's ascension as the Emperor of the Terran Empire.