LIKE JUST ABOUT EVERYTHING ELSE involving Marvel's mutant titles of the 1980s, a vast knowledge of the history of Marvel's mutants is necessary to understand what's happening.
The short version is this: Demons eager to extend their rule to Earth's dimension open a portal over (where else?) New York City. While the Marvel mutant teams fight back, the rest of Marvel's New York-based heroes deal with inanimate objects coming to life and other not-so-common events.
The long version is a little more complicated. The basis for this crossover was laid down in the 1983 Magik mini-series, which saw little Illyana Rasputin, sister to the X-Mens' Colossus, kidnapped by Belasco, ruler of Limbo. (Limbo is a dimension between dimensions; time has no meaning there, and the generally crummy real estate is made crummier by the hordes of demons and beasties.) She returned to Earth twice her age, having fought for her freedom by gaining control of the Soul Sword (and, hence, Limbo itself). She then joined the New Mutants as a member of the team, using her ability to travel to and from Limbo as a way to transport the team between places in the blink of an eye.
However, one transport went wrong, and Magik is forced to deal with N'astirh, one of Belasco's servants, to get her team home to Earth. Of course, a demon's word is never to be trusted, as he uses her to open a portal over New York City, allowing his armies to invade the city. Thousands of demons pour into New York's streets, and the Empire State Building becomes the focal point of their evil, gaining an extra 100 stories in the process. As New York's citizens flee in terror from the demons and rampaging mailboxes and taxicabs, N'astirh employs the powers of ten mutant babies to keep the portal open forever, but he's outfoxed by the X-Terminators, an even more junior team of mutants than the New Mutants. They succeed in damaging the portal, but not closing it.
Meanwhile, Madelyne Pryor, the wife of X-Man Scott "Cyclops" Summers, makes a deal with N'astirh. (At this point in X-history, X-Factor member Scott believes Madelyne and the X-Men are dead, since the world "saw" them die in the so-called "Fall of the Mutants." They're actually in Australia, using their "dead" status to stage covert operations.) Her son, Christopher, was kidnapped by a bad guy named (appropriately) Mr. Sinister, causing her to go mad. With a few goblins at her disposal, she calls herself the Goblin Queen, and uses her position inside the X-Men team to incite them against X-Factor, the mutant team composed of the original X-Men.
While the X-Men track down Sinister's Marauders (a team of mutant assassins under his command), Madelyne finds Sinister's lair, forcing him to tell the truth about her. Sinister reveals that she was a clone of Jean Grey, created to seduce Cyclops so that Sinister could have the son of that union, who promised to be a powerful mutant. Driven even more insane by this news, Madelyne takes her son and prepares to sacrifice him at the top of the Empire State Building to keep the gateway between Earth and Limbo open forever.
Of course, the X-Men defeat the Marauders, but Colossus leaves the team to find his sister, Illyana, who just happens to be fighting S'ym, another powerful Limbo demon, for control of the Soul Sword. She then throws the sword into the portal, sealing it forever and sweeping most of the demons out of New York City.
Meanwhile, X-Factor tries to save Nathan from Madelyne, but she had a plan prepared: Since the X-Men still trusts her, she pleads with them to keep X-Factor away from her son. Storm and Jean Grey figure out who the real enemy is, but their chat is interrupted by N'astirh, who isn't finished yet. The X-Men take care of him, and then turn their attention to Madelyne, who is dispatched in a psychic battle with Jean Grey.
The X-Men then learn that Sinister was behind Madelyne's actions -- not to mention her very existence -- so they track him back to Professor Xavier's mansion for a final showdown. They battle the Marauders and face Sinister himself, but he easily defeats them all, and tells Cyclops how he had manipulated his entire life in order to produce a child who would become the world's most powerful mutant. The X-Men retaliate, and Cyclops uses his eye-blast to reduce Sinister to bones, ending the Inferno at last.