Grue is a mysterious creature that never actually reveals itself. If the player is caught in complete darkness with no light source, the Grue will attack the player for 100 points of Health damage and 20 points of Sanity loss. This will continue until the player is dead or finds a light source.
Maintaining a source of light through the night will actually be the first challenge of the game (after not running into a dangerous creature) because starvation won't set in until the second night. Pretty much any light source at all is capable of keeping the Grue away including the faint light of a Crock Pot, Fireflies or the window light of a Pig's house. Keep in mind that some lights go out when approached such as the Pig's house so it is only good when "leap frogging" or running from one temporary light source to another.
The name Grue is actually taken from the source code of the game. The only other reference the Grue is actually from Maxwell himself who seems to recognize the Grue. Maxwell refers to the Grue by the name Charlie and it has been noted by the developers that Charlie is a girl. Charlie and Maxwell were friends though the Grue will still attack Maxwell at night when the player is playing as Maxwell.
The name, Grue, comes from the classic game Zork I. In this game, there was a similar creature that only attacked the player in complete darkness.