(covers information from several alternate timelines)
Ajur was a female Vorgon criminal from the 27th century who, along with her partner, 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")
Ajur was played by actress Karen Landry.
According to the script, Ajur's name was pronounced as "AY-zure". 
Ajur and Boratus shows up in the Star Trek Online Agents of Yesterday mission "Vorgon Conclusions". Here, we are shown their initial attempts to steal the Tox Uthat. However, instead of giving up at the end of "Captain's Holiday", they are tipped off that the Tox Uthat survived and was in Starfleet Headquarters. They arrive in 2375 during the Breen attack on Earth at the advice of the mysterious "Envoy". During the confrontation, Ajur is slain by the player character, leading to Boratus to ally with the Envoy in revenge.