Hakuryuu is the Sanzo Party's fifth member—basically the long-suffering pack mule carrying four loud guys around. He's a white dragon by default, though he can shift into a Jeep when the situation calls for it. Hakkai named him, but the name "Hakuryuu" stuck in the anime and most adaptations, partly because it was based on a trademarked product name. (In the manga, he's just called Jeep.)
He's an artificial creation, a experimental blend of science and youkai magic, which gave him those distinctive crimson eyes that match the dragons of Houtou Castle. Hakkai found him weakened in the forest while still living with Gojyo and brought him home. Since then, Hakuryuu's been more or less Hakkai's pet—though that relationship has evolved considerably over the series.
Hakuryuu and Gojyo didn't exactly hit it off at first, but they eventually made peace, partly because they're both "taboo existences" in their own ways. It's one of those dynamics where two outcasts just understand each other enough to coexist.
Content compiled by AnimeList.moe from publicly available sources.