Shinigami ("death god" or "death spirit") is an entity in Japanese folk lore that is responsible for seeing that a person dies at their appointed time and then guiding their spirit to be judged in the afterlife. In this respect, Shinigami are related to the Western concept of the Grim Reaper. Though the Grim Reaper is a single entity, there are multiple Shinigami which are often imagined as working together.
It is most commonly used in the Anime/Manga "Bleach" and the Anime/Manga "Death Note".
In Bleach, Shinigamis' (Soul Reapers in the English Translation) main jobs are to send spirits to the Soul Society (aka. the afterlife) and to cleanse Hollows.
In Death Note, Shinigamis are the (original) owners of notebooks called the Death Note. If a name is written in the Note, then the person of that name would die, and the responsible Shinigami would gain life.