According to the film's mythology, vampires drink the blood of humans to retain eternal life. However, only Miriam seems to enjoy eternal youth. The other vampires whom she creates eventually grow old, suffering from a sort of living death as their bodies can no longer sustain them. The film doesn't explain why Miriam remains young and the other vampires eventually grow old. The Strieber novel goes into more detail, saying that Miriam has never been human, and that she is likely the last of her species. Her formerly human consorts are thus vulnerable to aging in ways that she is not.
Miriam promises eternal life to those humans whom she feels are worthy. John began life as a cellist in 18th-century France before she came to him with an offer to become a vampire. When the film opens, they are living in modern-day New York, where they teach classical music while hunting for victims.
Miriam never told John the realities about his condition. As the film progresses, he begins to grow older and older, and feeding on blood no longer helps him stave off decrepitude. In desperation, he seeks out a doctor named Sarah (played by Susan Sarandon) who specializes in premature aging. She proves incapable of aiding him --- though she does draw Miriam's eye as her next potential mate --- and he eventually finds himself entombed in a coffin in Miriam's attic, joining other coffins containing all of her previous lovers.
At the film's climax, Sarah is transformed into a vampire, but refuses to succumb to Miriam's seductions. She stabs herself with Miriam's ankh/knife and forces Miriam to drink some of her own blood. Miriam takes Sarah up to the attic, planning to add her to the other withered lovers in their coffins. When she does, the other lovers --- including John --- break out of their coffins and push Miriam to her apparent death. When Miriam "dies," the lovers all collapse into dust, destroying them and ending their torment. (A final twist reveals that Miriam did not die, but rather transferred her eternal youth to Sarah. Sarah begins her new life as a vampire in Paris, with Miriam locked in a coffin in the basement.)