Ryuk (リューク, Ryūku) is a Shinigami who indirectly gives Light Yagami a Death Note. Ryuk was bored with the activities (or lack thereof) of the Shinigami Realm, and decided to obtain a second Death Note and dropping it in the Human World for someone to find, hoping to amuse himself from the boredom he had endured. This Death Note is found by Light, leading to his attempt to rid the world of evil, and easing Ryuk's boredom. He deliberately writes the instructions inside the front cover of the Death Note (in English, which he assumed to be the most popular language in the Human World) so people would understand its purpose. The Death Note is discovered by Light, and Ryuk follows him around for much of the series to see how Light uses it.
As Ryuk explains when he first meets Light, he will take Light's life when his time comes. In the manga, Ryuk does this after Light is shot several times by Matsuda. Light desperately begs Ryuk to write the names of the investigation team and the SPK members in the Death Note, Ryuk apparently starts writing them down, but Ryuk reveals he simply wrote Light's name instead. He was expecting Light might have thought of some way out of his situation, but seeing as he was desperate enough to go to Ryuk for help, he decided it was all over for him. Ryuk then returns to the Shinigami Realm after killing Light.
In the anime, Light does not ask for Ryuk's help and manages to escape the Warehouse thanks to Mikami's suicide distracting everyone else. However, his wounds are too severe for him to escape very far, and Ryuk, who is watching him from atop a pole, decides that following Light around during a prison sentence is not worth his time and thus writes Light's name in his Death Note. Light dies peacefully, with an apparition of L standing over him - mirroring Light standing over L on his death.