Khazad-dûm is the name the Dwarves gave to the underground city later known as Moria.
Khazad-dûm was the greatest Dwarven realm of the Misty Mountains. It was founded by Durin the Deathless beneath the mountain of Caradhras and grew into a vast network of halls, mines, and underground roads. The Dwarves discovered mithril there, making the kingdom exceptionally wealthy.
The Dwarves eventually delved too deeply and awakened a Balrog, which killed Durin VI in the Third Age. The surviving Dwarves abandoned the city, after which Orcs occupied it. By the time the Fellowship passes through, the former kingdom is widely called Moria, a Sindarin name associated with darkness.
Erebor is the Lonely Mountain, while Gondolin is a hidden Elven city of the First Age. Aglarond is another cavern complex connected with Helm’s Deep. These places belong to Tolkien’s wider geography but are not alternate names for Khazad-dûm.