Minas Morgul is a city on the continent of Middle-earth, in Arda, a fictional world originally created by J.R.R. Tolkien in the early twentieth century. It lay in the land of Imlad Morgul, at the western foot of the Ephel Dúath, the chain of mountains that acted as a natural border between the realms of Gondor (to the west) and Mordor (to the east). To its immediate east lay the pass of Cirith Ungol, a semi-secret way over the mountains and into the valleys of Mordor. To the south, Ithilien and the River Poros lay, and to the north was the Morannon (the Black Gate of Mordor), with the Dead Marshes beyond it.
Its name means "Tower of Black Sorcery" in the tongue of early Gondor, which is a variation of the Sindarin language spoken by most of the Elves at the time of the city's founding. The name seems to have been the same to speakers of Sindarin, Westron (the common speech), and the Black Speech of Mordor; all three languages referred to it as "Minas Morgul."
Originally, its name was Minas Ithil, and it was built by Gondor during the reign of Elendil, the first king and the founder of Gondor, to keep a watch over the border. The Gondorians built it in II 3320, and held it for nearly 2100 years afterwards. It was captured by the Nazgûl, the dark servants of Sauron, in the year III 2002, at which point it acquired its new name, as well as a vast number of new inhabitants, all of whom were greater or lesser servants of Sauron. As he grew in power toward the latter half of the Third Age, Sauron began building large armies and stationing them at Minas Morgul. He would use these armies to launch assaults on the nearby Gondorian cities of Osgiliath and Minas Tirith, and during the War of the Ring, it was his primary base of military operations (though he himself remained, semi-corporeal, within the tower of Barad-dûr in Mordor). During this time, Sauron's highest liutenant, the Witch-King of Angmar, ruled over Minas Morgul, so when he and the rest of the Nazgûl were roaming the lands and skies of Middle-earth in search of The One Ring, they came from, and returned to Minas Morgul in between searches.
In The Return of the King, Gollum/Sméagol leads Frodo Baggins, the Ringbearer, and his companion Samwise Gamgee past Minas Morgul and into Mordor, barely escaping the notice of the Nazgûl and any number of orcs stationed in the Tower of Morgul. As they pass by, they witness a huge garrison of Sauron's army issuing forth out of Minas Morgul, on their way to Minas Tirith to take part in the Battle of Pelennor Fields. Also in The Return of the King, the Captains of the West¹ and their army, having been victorious at Pelennor, march past Minas Morgul on their way to the Morannon, stopping briefly to deliver an announcement of defiance to the dark forces holding it.
In the movie version of the Lord of the Rings trilogy, we are given two glimpses of Minas Morgul; one brief long-shot of its gates, in the beginning of The Fellowship of the Ring, as the Nazgûl ride out in a fury after the last known whereabouts of The One Ring are discovered as Gollum/Sméagol is tortured at Barad-dûr. In The Return of the King, we're treated to a significantly more detailed (though still mostly obscured) view of the place, which seems very large and threatening. A creepy green light oozes from its tower, and an air of evil permeates the entire region, the white stone structures built by Elendil and his people defiled by Sauron's dark sorcery and the repugnance of the foul creatures inhabiting them.
After Sauron's armies were routed in the Battle of Pelennor Fields during the War of the Ring, no further aggression was launched from Minas Morgul. After the defeat of Sauron in III 3019, Gondor razed it completely, along with every other remaining place Sauron had corrupted, declaring that no evil thing shall be allowed to grow or be built where it had been ever again. The land it sat on, Imlad Morgul, thereafter became a part of Ithilien again, where it was ruled by Prince² Faramir and his wife, Éowyn, though it was wholly a part of Gondor.
1 - Aragorn, Elladan, Elrohir, Éomer, Gandalf, and Imrahil, among others.
2 - Faramir received the Princedom of Ithilien just prior to Aragorn's coronation, when he surrendered his Stewardship to the new King.