The bridge of the Dauntless in mid-transformation.
Particle synthesis was a technology which enabled the creation of artificial environments, in much the same way as holograms, but much more advanced and difficult to detect.
In 2374, Arturis utilized particle synthesis on the USS Dauntless, in an effort to fool the crew of the USS Voyager into believing that the vessel was of Starfleet origin. (VOY: "Hope and Fear")
In 2375, Species 8472 utilized particle synthesis in combination with holographic projections to recreate Starfleet Headquarters in their terraspheres, with which to train operatives to infiltrate Starfleet. (VOY: "In the Flesh")
Apocrypha
According to the Pocket Books novel Greater than the Sum, particle synthesis is related to the Quantum Slipstream technology. In the novel, it is explained that the particle synthesis works by exploiting, and altering, virtual particles on a quantum level.
Background
As a running gag, whenever a Voyager crewmember asked a being about this process, it was immediately dismissed as "Particle synthesis. Beyond your understanding". (VOY: "Hope and Fear", VOY: "In the Flesh") (citation needed • edit)