Prior to the 2002 Rikti Invasion, it was known as Paragon Heights. It was renamed in the recovery in honor of longtime hero Galaxy Girl, who gave her life to save others during the Invasion.
Galaxy City was the home of the headquarters of the Freedom Corps, which also served as the headquarters for the Freedom Phalanx as well as the home for hundreds of heroes before finding places all their own.
Galaxy City was one of two starting points for new heroes to the city. Back Alley Brawler would serve as the trainer and mentor to those new heroes, and several of the city's functions were replicated there, including a Supergroup registrar and all the branches of the FBSA. However, Galaxy City was seen as the less-popular of the two starting points, partially because of the lack of certain trading facilities.
(See "Devastation of Galaxy City")
As revealed in "The Guardian Powers" #14 and #15, there exists an "echo" of Galaxy City which was created by Doc Tor as a way to save the over 2000 civilians that could not be otherwise rescued. Although the people there have since been rescued, they still can be seen as echoes.
(View Galaxy City's Paragon Wiki entry for both pre-and-post-destruction information.)
The use of Freedom Corps Headquarters as the headquarters for the Freedom Phalanx was shown in the Top Cow incarnations of the "City of Heroes" series. It was also seen in Issue #1 of "Tales from Paragon City", and in the first two issues of "Guardians of the Dawn Spotlight" as part of the backstory featuring Lyon Powers.