The dark fortress where Sauron lives in Mordor is Barad-dûr.
Barad-dûr, whose name means “Dark Tower” in Tolkien’s Sindarin, is Sauron’s principal stronghold in the southeast of Middle-earth. It stands near Mount Doom, where the One Ring was forged and where it can be destroyed.
The fortress was originally built during the Second Age after Sauron began consolidating power in Mordor. It was destroyed at the end of the War of the Last Alliance when Sauron was defeated, but its foundations could not be completely removed while the One Ring endured.
Barad-dûr is not the same place as Minas Morgul, the former city of Minas Ithil, or Isengard, Saruman’s fortress in Rohan. Dol Guldur is another separate stronghold, located in southern Mirkwood.