Shinigami literaly translated means:
Death God, or
God of Death. 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.