Bellatrix, or Gamma Orionis, was a star with an associated system located in the Bellatrix sector of the Gamma Orionis sector block, in the Beta Quadrant in Federation space (STO video game: Star Trek Online, ST reference: Star Charts).
Bellatrix consisted of only a single B class blue star, which had a brightness of magnitude −3 (ST reference: Star Trek Maps, ST reference: Star Charts).
Bellatrix, along with Kessik and FGC-82659, was one of three points in space that defined the boundaries of the Delta Triangle (SCE eBook: Where Time Stands Still, ST reference: Star Charts).
The Federation starship USS Bellatrix was named for this location (ST reference: Star Fleet Technical Manual).
There was an Orion settlement on Bellatrix V (VOY video game: Elite Force).
The Bellatrix system was not a member of the Federation (ST reference: Star Charts).
Some information about the real-life characteristics of Gamma Orionis are derived from modern astronomy, as described on the Wikipedia page for this topic. It has the stellar classification B2III, and is visible from Earth in the constellation of Orion, 240 light-years distant.
In VOY episode: "Relativity", Kathryn Janeway identifies "Bellatrix" as an Arabic name, although the word is actually Latin.