The Shinigami King presides over all human life. He is the one who delegates lifespans to humans and Death Notes to shinigami. The reasons behind this are unknown and next to nothing is really known about this entity. The way the shinigami talk about him, he is apparently their boss but it is unknown how they serve him as the only reason they kill humans is for their own survival.
While frequently mentioned, the Shinigami King does not physically appear in the main Death Note story. His physical form is however shown in a bonus story as Ryuk mentions how he's started a trade service. He'll allow the shinigami an extra Death Note to give to humans in return for apples, which he has since become addicted to similar to Ryuk.