(covers information from several alternate timelines)
Doctor Leah Brahms was a member of the Theoretical Propulsion Group and a graduate of the Daystrom Institute of Technology. She made major contributions to the development of the Template:ShipClass's warp drive system, far beyond her official role as a junior team member.
In 2366, the USS Enterprise-D was caught in an ancient trap which drained the energy from its engines and bombarded the crew with radiation. To learn more about the engine design and help him find a solution to the problem, Geordi La Forge created a holographic version of Dr. Brahms based on her Starfleet personality profile analysis from stardate 40056 (2363) and her public persona as recorded from her appearances at the Chaya VII Intergalactic Caucuses. The holographic Dr. Brahms was opinionated on engineering matters but still friendly, and La Forge developed an attraction to her image. Together, they were able to save the Enterprise, and La Forge shared a kiss with the hologram before ending the program. (TNG: "Booby Trap")
A year later, in 2367, the real Dr. Brahms, by that time promoted to Senior Design Engineer of the Theoretical Propulsion Group, visited the Enterprise for an inspection of the field modifications made to the ship's engines by Commander La Forge. Dr. Brahms was far different from her hologram; she was cold, all business and without humor. Much to his dismay, she was highly critical of his work, especially his enhancements of the warp engines. La Forge and Brahms tried to work out their differences and got along until Brahms informed him that she was married, much to Geordi's shock and dismay. She later discovered the records of her hologram's somewhat romantic interactions with Geordi during his attempts to resolve the aceton assimilator crisis and strongly objected to his having programmed a holographic replica of herself. She told him that she felt violated, to which Geordi expressed his indignation of her judgmental attitude. The two engineers then pulled together in a crisis and became good friends. (TNG: "Galaxy's Child")
In 2370, in an alternate timeline, where Harry Kim and Tom Paris didn't join the crew of the USS Voyager, Leah Brahms had become the director of the Zefram Cochrane Institute for Advanced Theoretical Physics. She signed her name onto the Cochrane Medal of Excellence for Harry Kim on stardate 47302.5. (VOY: "Non Sequitur", production art )
In an alternate future around 2395, Dr. Brahms was married to La Forge. The couple were living on Rigel III and they had three children together: Alandra, Bret and Sidney. During this time, Leah had become the director of the Daystrom Institute, as well as quite a gardener. (TNG: "All Good Things...")
According to the Star Trek Encyclopedia, "In an early draft of "Booby Trap", Brahms was named Navid Daystrom, presumably a descendant of Doctor Richard Daystrom. Unfortunately, the casting department did not realize that this would require a black actress to play the part until after Susan Gibney had been hired. At the suggestion of script coordinator Eric Stillwell, the character was renamed, but the Daystrom tie-in was kept by adding a line that she had graduated from the Daystrom Institute."
According to a line in the revised final draft of "Galaxy's Child" , Brahms' husband's name was Michael, and he was an engineer on Garran IV.
In the original script for Star Trek Nemesis , Leah and Geordi had in fact become a couple by 2379, and the two of them attended Will Riker and Deanna Troi's wedding together. However, because actress Susan Gibney was unavailable, her character was removed, and several of her lines were instead given to Guinan.
In the Genesis Wave series, set in 2377, Leah Brahms is the first person to discover that the Genesis Device technology has been stolen, surviving the initial destruction caused by the wave thanks to her experiments with phase-shifting technology as a suit (Although her husband is killed). During the crisis, she is given command of a Klingon ship, with Maltz, one of the few survivors of the original Genesis crisis, serving as her first officer due to the respect he develops for her during their attempts to escape the Genesis Wave. Thanks to her research, the location of the source of the Genesis Wave is discovered, with Maltz destroying it on a suicide mission. In a subsequent conversation with Geordi La Forge, he admits his love for Brahms, but assures her that he expects nothing in return. La Forge only asks her to meet him at an engineering conference in a few months' time, and she agrees.
In the novel Indistinguishable from Magic, set between 2382 and 2383 Brahms is reunited with the newly promoted Captain La Forge during the U.S.S. Challenger's assignment to investigate trans-slipstream drive. The two begin a romantic relationship, and after the Challenger is destroyed, Brahms moves in with La Forge aboard the Enterprise-E after he accepts his original position. She is convinced to join him when he offers her the opportunity to work on only one vessel, instead of many.
The storyline in the Star Trek: Countdown comic indicates that La Forge and Brahms are married by 2387.