In the 27th century, two Vorgon criminals, Ajur and Boratus, attempted to steal a device known as the "Tox Uthat", a quantum phase inhibitor capable of halting all nuclear fusion within a star. In order to prevent the Uthat from falling into the wrong hands, Kal Dano, the Uthat's inventor, traveled back in time to the 22nd century and hid it on Risa.
Having failed to obtain the Uthat, Ajur and Boratus searched historical records and discovered that Jean-Luc Picard would discover and destroy it in 2366. Traveling back to that point in time, they told Picard that they were security agents and that they wished to retrieve the Uthat for safekeeping.
When Picard learned, however, that it was two Vorgons who attempted to steal the Uthat in the first place, he destroyed it using Transporter Code 14, thus fulfilling his role in history. Failing in their quest to obtain the Uthat, Ajur and Boratus returned to their own time. (TNG: "Captain's Holiday")
The script for "Captain's Holiday" describes Ajur and Boratus as simply, "Two sleek and well dressed aliens". 
The inspiration for this makeup design was "created from the look of a scallopy seashell, the Vorgons had crests that ran up their faces and all the way to the backs of their heads. There was one prepainted headpiece and additional appliances for the face and the throat. After all the appliances were attached, the makeup artists airbrushed color to match and then applied a coat of iridescent paint to create a glow on camera. Each Vorgon also wore an electronic earring that had a random pattern of blinking shapes and gave the effect that these creatures were indeed time-travelers from the twenty-seventh century." (Star Trek: Aliens & Artifacts, p.118)
In his review of "Captain's Holiday", Star Trek author Keith R.A. DeCandido expressed his distaste at the Vorgons in the episode, writing "Karen Landry and Michael Champion are pretty awful as the Vorgons, though I suspect they were having trouble even talking under that silly makeup." 
The Vorgon costume as worn by Karen Landry as Ajur was later sold off in the It's A Wrap! sale and auction for US$312.00. It comprised a purple Lycra jumpsuit with metallic purple and silver crinkled fabric at the knees and elbows, along with long neoprene shorts, an accompanying crop top and matching knee-high boots and gauntlets. 
According to Q's Guide to the Continuum, the Vorgons would not have been a threat even if they had gotten their hands on the device as they would have destroyed themselves with it anyway.
The Vorgons are also featured in the Star Trek Customizable Card Game in the form of a "Vorgon Raiders" card depicting Ajur and Boratus.
According to Decipher's Aliens sourcebook, the Vorgon homeworld was called Vorgos and was located in the Delta Quadrant.
Ajur and Boratus appear in the Star Trek Online "Agents of Yesterday" expansion. The mission "Vorgon Conclusions" details their attempts to steal the Tox Uthat following the events of "Captain's Holiday". Ajur and Boratus are tipped off by the mysterious "Envoy" - leader of a Temporal Cold War faction opposed to the Temporal Accord – that the Tox Uthat survived its apparent destruction by Captain Picard, and was hidden in Starfleet Headquarters during the Dominion War. They travel to 2375 and attempt to infiltrate the Tox Uthat's storage place under cover of the Breen's surprise attack on Earth, only to be intercepted by the player character and agent Daniels. During the resulting firefight, Ajur is killed, and Boratus swears vengeance, prompting the Vorgon to join on the side of the Envoy, who leads a coalition of Krenim, Na'kuhl, Vorgon, and Sphere-Builder extremists against the Federation and its allies at the Battle of Procyon V.