Though the Gelth originally had biological bodies, they were eventually reduced to a gaseous form during the Time War, after their bodies were destroyed during the conflict. They could shape this gas into a wraith-like form, but they needed to be in a gaseous environment to survive. Once there, the Gelth emitted ghostly screams as they floated around the location at which they congregated.
The Gelth could take control of human corpses, though only with difficulty. While inhabiting the corpses, the Gelth could control and move the dead humans as walking zombies, leaking glowing blue gas that streamed through the mouth. While controlling a human, they tended to continue unfinished business. The Gelth would eventually be forced out of the corpse over time or if there was a lot of gas in the air around them. (TV: The Unquiet Dead)
During the Last Great Time War, billions of Gelth were turned into gas. Eventually, they discovered the Rift, which opened into Gabriel Sneed's mortuary in 1869. They were attracted to the area, since they could inhabit the gas found in the pipes and emerge through gas lamps. They also discovered that they could survive in the gases in a decomposing human and, as the Ninth Doctor put it, "use it as a vehicle". The Gelth contacted Gwyneth, Sneed's servant, who had psychic ability. She thought of the Gelth as angels. Through Gwyneth, the Gelth spoke with the Doctor.
Presenting themselves as peaceful and facing extinction, the Gelth asked for the Doctor's help in allowing Gwyneth to open the time rift further, allowing through more of their kind. The suggestion of doing this and allowing more Gelth through the rift to inhabit the corpses of humans was an idea which horrified the Doctor's companion, Rose. The Doctor, however, accepted the idea as an interim stage before taking the Gelth to a place where he could help them acquire new, specially-created bodies. With the rift opened wider, the Gelth, however, planned to let all the billions of Gelth through and take the planet by force. Gwyneth closed the rift in time, however, and destroyed the remaining Gelth with fire. (TV: The Unquiet Dead)
When travelling alone, the Tenth Doctor encountered a small group of Gelth in the crypt of Cheltnam Abbey who bridged a gap in the Rift through Preston Gilchrist while on the TV show Scared Stiff. The Gelth possessed the bodies of several skeletons from the crypt of the abbey, along with Preston, as they attempted to open the Rift permanently. The Doctor distracted Preston by revealing that the television show was a fake, thereby giving him time to close the Rift and trap the Gelth. (PROSE: Scared Stiff)
While attempting to bluff the Sycorax into leaving Earth alone, Rose tried to invoke the authority of the "Gelth Confederacy", among other species she had encountered on her travels. (TV: The Christmas Invasion)
Behind the scenes
According to The Brilliant Book 2012, a book that contains non-narrative based information this wiki doesn't consider a valid source, in the timeline created by River Song refusing to kill the Doctor, the Gelth tried to contact Charles Dickens via social networking websites. Dickens told them he wasn't "falling for that one again" and blocked them.