The Orion system was an inhabited star system in the Orion sector of the Alpha Quadrant. The primary was named Orion. (TOS: "What Are Little Girls Made Of?": TAS: "Yesteryear"; Star Trek VI: The Undiscovered Country, production art; VOY: "Good Shepherd")
In "The Explored Galaxy" star charts, Orion was located between the Tholian Assembly and Romulus. Both were identified in DS9: "Call to Arms" as being located in the Alpha Quadrant. According to The Worlds of the Federation (page 36), the Rigel system and the Orion system are one and the same. The eight planet in the system is the planet Orion, the homeworld of the Orion species.
According to Star Trek: Star Charts (pgs. 36, 45, 60), the Orion system was Pi 3 Orionis, a trinary star system located in the Beta Quadrant. The primary was a Class F star with a magnitude of +3, which was ten times the brightness of Sol. Secondary and tertiary were Class F stars. In 2120, the system was a destination on the final voyage of Zefram Cochrane. In the mid-22nd century, the system was in non-aligned space, and was in proximity of the Vulcan trade routes. In the late 24th century, the system was located in or near Federation space, and it was a destination on the major space lanes.