The film opens with a bizarre sequence where we see an unidentified individual, visible only by his hands, creating a strange weapon: a glove with sharp knives for fingers. Tina, a young girl in a nightgown, is rushing down ominous alleyways filled with steam and shadows. Suddenly the bizarre man lunges at her, brandishing the glove.
Tina screams and sits bolt upright in her bed; the entire episode was a nightmare. Tina's mother comes into her bedroom, alerted by her scream, and notices that Tina's nightgown is slashed near her stomach. The cuts are similar to what would be made with the glove from the dream.
Tina meets her friends at school the next morning.
Her boyfriend Rod, her best friend Nancy, and Nancy's boyfriend, Glen.
That night, Nancy and Glen decide to stay with Tina, because her mother has gone out of town. Rod shows up later and frightens the three of them by making strange sounds in the back yard and then jumping out at Glen.
Nancy and Tina discover that they have both had the same nightmare, and although Glen and Rod try to hide the fact, they seem to have been having nightmares, too. Rod and Tina go upstairs into a bedroom to have sex, while Nancy and Glen sleep separately downstairs; Nancy resists the idea of fooling around, since she and Glen are supposedly there as moral support for Tina.
As all of them fall asleep, strange things begin to occur. The wall over Nancy's bed seems to lose its solid form, and the shape of a human figure looking at her seems to lean out of it. Nancy does not see it, her eyes closed in half-sleep, and when she awakens, the form disappears.
In the room where Tina and Rod are sleeping, Tina is awakened by noises at the window; someone is in the back yard hurling small rocks at the glass.
Tina knows this is whoever has been trying to frighten her, and she goes downstairs to confront whoever it is.
When she ventures out into the back yard, a phantom voice lures her into the alley, the same as in her dream. She is startled by strange things such as a lamb suddenly wandering into the alley.
Finally, her stalker emerges, a man in a fedora, a dirty red and green striped sweater, and the deadly glove with knives for fingers.
He taunts Tina by laughing at her and whispering her name, and he walks towards her with his arms stretching further than humanly possible, spanning the entire alley, blocking her escape.
Tina retreats into her back yard and the figure leaps out again, this time from behind a tree.
He slices off his own fingers and strangely colored blood spurts out. Tina tries to make it into her back door, but the man grabs her, and she struggles with him. We see Tina in bed next to Rod, trashing around; she has been dreaming this incident, as well, only the man seems to actually be under the covers with her.
Rod awakens and pulls off the sheet. He can only see Tina, but she is being manipulated by some unseen force. Her stomach is slashed open, four wounds appearing all at the same time, obviously from the glove. Tina is dragged screaming up the wall, over the ceiling, and falls down dead into the bed, which is now covered in her blood. Nancy and Glen have now awakened and burst into the room to find Rod gone, apparently through the window. Tina is dead.
At the police station, Nancy is giving her account of what happened and she talks to Lt. Donald Thompson, who also happens to be her father. Nancy's mother, Marge, and Lt. Thompson are apparently divorced, and they greet each other with restrained hostility.
Rod is the obvious suspect in the murder, but Nancy feels that this is not the case; although Rod has a reputation as a delinquent, Nancy knows he is not a killer. As Nancy walks to school the next morning, Rod appears and pulls her into some bushes.
Rod swears to Nancy that he did not kill Tina, but before they can talk any further, Lt. Thompson appears with his men and arrests Rod.
Nancy is offended that her father used her to capture Rod, and she storms off.
At school, the exhausted Nancy falls asleep in English class, and she thinks she sees Tina standing outside the classroom door. Tina speaks to her from a bloody body bag, and Nancy sees the bag dragged away by some unseen hand. She drifts out of the classroom and follows the bloody trail down into the basement.
On the way, she is confronted by a female hall monitor, who scolds her for not having a pass. Leaving the girl behind, Nancy turns around when she speaks to Nancy in a man's voice, brandishing the weird knife-glove and cackling at her.
Nancy goes down into the basement and is confronted by the killer, who is dressed in the same sweater and hat.
His skin is horribly scarred, burned almost beyond recognition.
He traps Nancy in the boiler room and backs her against the wall, taunting her. "Come to Freddy!" he says, but just before he can slash her with the glove, Nancy burns her arm on a hot pipe. The burn wakes her up, and she disrupts her English class by screaming and jumping out of her chair. Even though she is now awake, she has a large burn mark on her arm. Embarassed and frightened, she leaves school for the day.
She visits Rod in prison and confirms her fears: Rod is having the same nightmare, haunted by the same figure with the glove.
She tells Rod that she knows he didn't kill Tina, and leaves.
Nancy has another incident where she is threatened by "Freddy"; she falls asleep in the bathtub and she is pulled underwater, dragged into some kind of abyss underneath.
She manages to swim back up to the opening of her bathtub and pulls herself to safety. Marge, hearing Nancy's screaming, bursts into the bathroom, but Nancy tries to pretend she simply fell.