Warning: Some spoilers for Stargate SG-1 are contained herein.
In Stargate SG-1, Naquadria is a variant of naquaada/naquadah that is far more powerful, but also far more unstable and dangerous.
It is believed that the Goa'uld system lord Thanos created naquadria roughly three millennia ago on the planet of Langara, through a process of exposing naquadah to a strictly controlled explosion. He nearly destroyed the planet in the process. Other Goa'uld remained unaware of the element until Anubis probed the mind of Jonas Quinn and discovered its existence.
Discovered later by citizens of the Langaran country of Kelowna, the Kelownans began experimenting with the element in an effort to create a weapon to use against other nations on their planet. Jonas Quinn was a member of the team of scientists working on the weapon, and was saved along with the rest of his team by Daniel Jackson when an uncontrolled reaction began building up during an experiment. Jackson removed the core sample from the experimental chamber manually, exposing himself to a lethal amount of radiation in the process. To repay Daniel and his team for what he did, as well as due to moral objections to his country's plans to develop a naquadria-based bomb, Jonas stole a significant quantity of the element and escaped through the Stargate. This branded him a traitor in the eyes of the Kelownans.
The Kelownans finally managed to create a massive naquadria bomb, which they used against their rival countries. This caused untold devestation that nobody was prepared for, and succeeded in ending a short planetary war in much the same way the atomic bomb ended World War II.
Naquadria is used by the Tauri (us) to develop the hyperdrive engines used in the X-302 and X-303, but is found to be too unstable. It turns out the naquadria's instability increases geometrically in proportion to the distance travelled through hyperspace. Thus, the Tauri ships are limited to short bursts through hyperspace, maxing out at a fraction of a light year.
Aside from being extremely energetic and unstable, naquadria has been shown to cause symptoms resembling advanced schizophrenia in humans and Langarans. All of the scientists involved in the initial naquadria research eventually succumbed to these symptoms, with the exception of Jonas Quinn, who did not work as long with the element as the others and thus remained unaffected.
When Anubis learned of the existence of naquadria, he travelled to Langara in an attempt to capture their stockpile and enslave the Langaran people. SG-1 used this event to stage an ambush in concert with the other Goa'uld system lords, destroying Anubis' mothership. Anubis himself, however, escaped.