The vast Kriosian Empire was once ruled by two brothers, named Krios and Valt, from the ancient Temple of Akadar on Valt Minor. They became divided over the love for a woman, an empathic metamorph known as Garuth. They both loved Garuth with such passion that the empire fell. Krios eventually kidnapped Garuth from their homeworld and took her to the planet that became his namesake. This action started a centuries-long war between the two worlds and subdivided the species into the Valtese and the Kriosians. (TNG: "The Perfect Mate")
By the 22nd century, the Kriosians were under the rule of the Sovereign Dynasty of Krios Prime, a family known on hundreds of worlds. The royal guardsmen of the Dynasty were known as the Sovereign Guard. Their leader was known as the First Monarch.
In September 2152, the future First Monarch Kaitaama was returning from a diplomatic mission when the Retellians attacked her transport. They killed her guards, placed her in stasis, and attempted to hold her for ransom. En route to their destination, the stasis pod in which she was held aboard the kidnappers' ship began to malfunction.
They were assisted by Enterprise, which later discovered the truth behind the kidnapping when their own chief engineer, Charles Tucker III, was himself kidnapped. Both Tucker and Kaitaama were later rescued, and the First Monarch was transported back to her homeworld. Kaitaama later ascended to the throne around May 16, 2153. (ENT: "Precious Cargo")
In 2367, the Kriosians began fighting for independence from the Klingon Empire. During the revolt, the rebels hid in the Ikalian asteroid belt, where they used the actinides in the asteroids to hide from sensor scans. The rebels later attacked a Ferengi freighter and a Cardassian freighter, in the belt.
For the most part, the Klingons tolerated the rebellion, until Vagh, the Klingon governor of Krios, charged the Federation with supplying the rebels with phaser rifles. Upon further investigation by Starfleet, it was discovered that the suppliers were, in fact, the Romulans, who were planting the evidence in attempt to drive a wedge between the Klingons and the Federation, and to jeopardize the Treaty of Alliance. (TNG: "The Mind's Eye")
The following year, Krios and Valt Minor finally agreed to a Ceremony of Reconciliation to finally bring an end to their centuries-long war. The Ceremony was held aboard the Federation starship USS Enterprise-D, at a point midway between the two systems.
Kamala, the first female metamorph born on Krios since the mid-23rd century, was bred for mating to Chancellor Alrik, the leader of Valt Minor. Leaders of both worlds, in the hopes of restoring peace, prearranged this event at the time of her birth. Kept in stasis during the journey, Kamala was the key to peace between the two worlds.
The Ferengi became involved when they learned of the treaty, attempting, in their own right, to acquire Kamala. They approached Kriosian ambassador Briam, feigning the wish to have exclusive rights to transport all Kriosian products to the Valt system. Eventually, their involvement led to the premature waking of Kamala, and a near-fatal accident with Briam. Following the accident, Jean-Luc Picard was chosen to negotiate in his place in an attempt to represent Kriosian interests of peace.
Despite receiving the gift of Kamala, Alrik was more concerned about the remaining provisions in the treaty, including unacceptable aspects of proposed trade agreements, the review of uncommitted territories, and the schedule for exchange of technology.
Ultimately, the negotiations and preceding Ceremony were successful. The wedding, held in a holodeck recreation of the Temple of Akadar, went as planned, as Krios and Valt Minor took their first steps towards peace. (TNG: "The Perfect Mate")
Physically, Kriosians were almost identical to their ancestors, the Valtese, possessing a smattering of spots that led from the top of their forehead along the side of their face to their back, coming to a point about halfway down their spine.
Kriosians were also known for their somewhat randomly-born empathic metamorphs. While male metamorphs were somewhat common, female metamorphs only appeared once every seven generations and were highly sought after as mates. These metamorphs also experienced the Finiis'ral, a life stage similar to late-stage Human puberty where they produced heightened pheromones. During its end, a metamorph would morph and imprint for the final time, a process that took about two days as the metamorph reached absolute sexual maturity.
Society and culture
The Kriosians had a structured class system, divided between the nobility and the commoners.
Kriosian culture during the 22nd century did not allow the First Monarch to socialize with the opposite sex. If anyone attempted to "make a pass" at the Monarch, the Sovereign Guard would cut off one of their hands. (ENT: "Precious Cargo")
Female metamorphs were commonly taken from their parents by the age of four and placed in a royal village where they were prepared by their tutors and servants. These attendants would constantly educate the metamorphs in literature, history, art, and sex. (TNG: "The Perfect Mate")
In the third draft script of "The Perfect Mate", this species was called "Krians". However, this was changed to "Kriosians" by the time the final draft of that script was issued. Both drafts of the script described the species as having "dark skin and hair, much in the manner of those from Earth's Indian subcontinent." 
In to an internal reference document, listing all the aliens seen in Star Trek: The Next Generation, which was circulated around the time of production on TNG Season 5, the Kriosians were described as "tall and slender, dark-spotted skin pattern along the hairline and down the neck and spine." 
Since the pronunciation guide in the archival final draft script of "Precious Cargo" refers to the language used by Kaitaama as "Tiburrian", it's possible she and the other Kriosians in that episode were at one point to have instead been called "Tiburrians". "Tiburrian" may alternatively refer to the location on Krios from which they hail or a language common to the people of that location.