(covers information from several alternate timelines)
Doctor Leah Susan Brahms was a female Human scientist of the 24th century United Federation of Planets. Considered a leader in warp field theory and its applications, she made major contributions to the development of the Galaxy-class' warp drive system. She became a professor at the Daystrom Institute.
Brahms attended the University of Alpha Centauri where she earned a Bachelor of Science in Subspace Physics and graduated cum laude. After graduation, she attended the University of Tomobiki where she wrote the masters' thesis "Subspace Processing Modes in Warp Propulsion Applications" and was awarded a Master of Cybernetics in Optical Technologies. Finally, she attended the Daystrom Institute of Technology where she wrote the doctoral thesis "Higher Order Warp Field Propulsion Applications" and was awarded a Doctor of Theoretical Physics.
Brahms started her career as a research assistant for the Daystrom Institute, developing protocols for measuring higher order subspace field distortions. During her tenure at the Daystrom Institute, she wrote articles that were published in the Scientific Tasmanian. She was later promoted to junior engineer and was assigned to Design Team 7 at the Utopia Planitia Fleet Yards; she was then promoted to team leader of this team. During her time with the design team, she was credited for improved dilithium crystal servo subsystems and a plasma resonance sampling device. Next, she was assigned to lead the Theoretical Propulsion Project, on which she was promoted to design engineer. During her time as a design engineer, she supervised the implementation of improved subspace field generators onto the Galaxy-class USS Enterprise-D and USS Yamato. After this, she was named a Starfleet Design Consultant.
Brahms became a Professor of Theoretical Physics at the Daystrom Institute of Technology on Mars, where she was awarded an honorary Feynman Chair. She lived at Quayle Canals Northeast, Utopia Planitia Gardens. (TNG: "Booby Trap")
Inspection of the Enterprise
In 2366, the 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")
Since the Enterprise-D was commissioned, Dr. Brahms learned of the field modifications La Forge had made to the ship's engines. She was critical of his work, but did not reach out to him. Finally, Starfleet Command had her go to the Enterprise-D to inspect his designs in 2367. She boarded via Starbase 313 soon after stardate 44614.6. Being protective of her designs, she treated him with disdain and didn't respond to his friendly overtone. Furthermore, she picked up on his over-familiar gestures and asked him about it, but he said he was just a fan of hers. She also replied she was married, which surprised him. La Forge and Brahms tried to work out their differences until she discovered the records of her hologram's somewhat romantic interactions with La Forge 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 La Forge expressed his indignation of her judgmental attitude. The two engineers then pulled together in a crisis when the Enterprise encountered a spaceborn creature whose offspring had drawn energy from it. Afterwards, they became good friends. (TNG: "Galaxy's Child", "Relics")
Later in life
In 2370, Leah Brahms was 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, La Forge was married to someone named Leah, possibly Dr. Brahms. The couple was 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...")
- Operating Protocol - Subspace Field Generators
- Operating Protocol - Subspace Harmonic Monitors
- Operating Protocol - Primary Field Coils
- Operating Protocol - Secondary Field Coils
- Subspace Design Logs - Theoretical Propulsion
- Gravimetric Power Limits (Theoretical)
- Emergency Plasma Vent Procedures
- Synthetic Grav Field Interaction Considerations
- Emergency Field Quench Procedures
- Field Maintenance Power Requirements
- Secondary Harmonic Safety Requirements
About the character's creation, the Star Trek Encyclopedia (4th ed., vol. 1, p. 100) stated, "In an early draft of "Booby Trap", Brahms was named Navid Daystrom, presumably a descendant of Dr. 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."
In the original script for Star Trek Nemesis, Brahms and La Forge 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 the scene with La Forge was re-written for Guinan. 
The All Good Things... novelization made it clear that the "Leah" referenced in the episode was indeed Brahms.
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 USS 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 USS 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. Subsequent novels taking place over the next two years,such as the Cold Equations trilogy and Takedown, depicted La Forge in his relationship with Tamala Harstad, a doctor aboard the Enterprise, with no mention of Brahms until The Light Fantastic when La Forge was on sabbatical in San Francisco and the two were obviously enjoying each other's company, despite Geordi's misgiving about having been "seeing someone else" before he left the Enterprise. Once Geordi returned to the ship, all subsequent novels show him and Harstad as a couple, again with no mention of Brahms, until Collateral Damage when Brahms invites the two of them to dinner in Berlin, and La Forge expresses worry that the three of them socializing could turn his open relationships with both them (of which both women were aware and accepting) awkward. Brahms suggested he ask Harstad and see what she thought.
The storyline in the Star Trek: Countdown comic indicates that La Forge and Brahms were married by 2387.