When is a PET scan useful?

Most PET scanners are combined with CT scanners, as in this PET-CT machine, in order to provide hybrid imaging that utilises the best of both types of imaging. — Filepic

Not to be confused with creatures that we keep as pets, the similarly-named PET scan is an imaging modality used in the medical field.

What is a PET scan (or simply PET) and why is it necessary?