By many, they are named Moria.
The Dwarves name it Khazad-Dum. In the mines of Khazad-Dum, there is the dwarf-city of Dwarrodelf (Elvish, or Mannish name).
Dwarrowdelf IS Khazad-dûm, even with the same meaning; in the grey-elven tongue "Hadhodrond"/"Nornhabar", highelvish "Casarrondo", but the elves later - after the Balrog appeared - called it Moria which means something like "Dark Chasm" (or Black Pit, Black Gulf). Westron name: Phurunargian.
Other "mannish" (translated) names - Dwarrowmine, Dwarrowvault
Didn't we have one of those lovely collective Question-and-Answer-threads somewhere? *searches*
__________________ Life is complex: it has both real and imaginary components.