The Redhorn Pass attempted by the Nine Walkers on their quest for Mount Doom lay beneath its slopes, linking the former Noldorin realm of Eregion in the west to the Dimrill Dale and hence the Vale of Anduin in the east.
After the fall of Khazad Dum, this pass was predominantly used by elves travelling between Lorien and Eriador.
Caradhras (in Khuzdul, the language of the Dwarves, Baranzibar) was one of the Mountains of Moria, the three mountains which the great Dwarf palace of Khazad-dûm was built under. Caradhras was also the site on Middle-earth where Mithril was found. It was here, though, that miners found Durin's Bane, the Balrog of Moria.
Caradhras was called the Cruel by the Dwarves and had long had a bad reputation. The Redhorn Pass (also called the Redhorn Gate) was known to be treacherous. It was on this pass that Celebrian, the wife of Elrond, was captured by orcs. This was also the pass used by the hobbits migrating from Gladden Fields into Eriador.
In Peter Jackson's Lord of the Rings movie, when the Fellowship is trying to cross Caradhras, Saruman conjures a spell that causes violent snowstorms on the mountain. In the book, the Company generally blames the mountain itself.