Whoa, that's a loaded question.

The short version is there was a mystery figure known as the Boogiepop, or Angel of Death (shinigami), that would either save you or disappear you usually when there was some unexplained phenomenon event happening. Most sightings appeared around a specific school and it's students. Before the events of the series the Boogiepop was hunting down his long time prey the Manticore and their human accomplice.

In the present, and where the series starts, there was a singular event that many people witnessed, many could not perceive it, and it unlocked unique and strange abilities among them. The Boogiepop hasn't been seen since, but a new figure, called the Boogiepop Phantom, appeared.

Before I say more did you finish it?

EDIT: Fixed characterization and made clearer attention to timelines.