Carn Dûm is very difficult to translate. Indeed, not only is there no attested meaning of the name, but we can't even be sure of the language it comes from! There are at least four possibilities.
The most obvious relies on the Elvish word car(a)n, meaning 'red'. From this, many have assumed that the entire name is Elvish, and presumably Sindarin. In support of this, the name Angmar, for the land where Carn Dûm stood, seems to be definitely Sindarin. On the other hand, there is no known connection between Carn Dûm and the colour red, and - a serious difficulty - dûm is almost certainly not an Elvish word.
Dûm isn't Elvish, but it is one of the very few words of Dwarvish vocabulary that we know for certain. In Khazad-dûm, it means 'halls, mansions', and this fits neatly for a mountain fortress or citadel. What's more, nearby Mount Gundabad has Dwarvish associations, so the possibility arises that Carn Dûm might have once been a city of the Dwarves, named in their language. There's no direct evidence to back this up, though, and carn doesn't seem to be a Dwarvish word.
The name might conceivably represent a compound of Elvish and Dwarvish, with a meaning something like 'red mansions'. However, it should be noted that these two languages are almost never seen in combination. Probably the only definite case is Finrod Felagund, for which the historical origins are well understood. We have no such explanation for Carn Dûm, though a compound origin remains at least a possibility.
The possibility that Carn Dûm comes from the Black Speech must also be considered, but we simply do not have sufficient information on this language to make a useful judgement.
A final possibility is that Carn Dûm comes from a Mannish language. Tolkien used real languages to represent the Mannish languages in his books, and especially Old English and Old Norse. Carn Dûm doesn't seem to belong to either of these, but there is another candidate: carn dúm are words from Gaelic that can be translated 'mountain fortress'. Did Tolkien intend this, or is it a spectacular coincidence? To accept it as intentional, we'd need to assume an entire new 'Angmarian' language, based on Gaelic, that was wiped out with Angmar by the Gondorians and survived only in this one name. This seems unlikely in the extreme, but where Tolkien is concerned, anything is possible...