- "You have to go after him."
- ―Jojen Reed to Bran Stark
The three-eyed raven is a figure that appears in Bran Stark's dreams, following his fall and injury. In Bran's dreams, the raven appears to be trying to lead him into the Stark family crypt, predicting his father's death.
The raven appears to Bran Stark multiple times in his dreams. It appears as Bran is about to fire an arrow and causes him to stop before he fires. The raven lands on the head of a direwolf statue and cries once, and then flies towards the Stark family crypt.
Bran dreams once again of the three-eyed raven. In the dream he's able to walk and attempts to shoot at him with a bow and arrow. Jojen Reed then appears in the dream, and tells Bran that he cannot kill the raven, because Bran is the raven. Later Jojen and his sister Meera encounter Bran and his companions in real life, and Jojen explains the prophetic powers of greensight dreams to Bran. Jojen explains that he also saw the three-eyed raven and it was Jojen himself in the dream (not just a prophetic vision of him), as the power of greensight allowed his mind to enter Bran's dreams.
When Bran touches a weirwood in the haunted forests beyond The Wall, he sees flashes of the three-eyed raven from his dreams at Winterfell, flying through the crypts. Among his other vision is a single giant weirwood on a hill, with a voice that whispers, "Look for me... beneath the tree. North." 
Behind the scenes
The overall design of the three-eyed raven was developed by William Simpson. There was considerable deliberation on where exactly the third eye should be located, as it wasn't specified in the books. For a time, Simpson considered actually putting it in the back of the head, to give it a full 360 degree field of vision. However, he later settled on putting it in the middle of the forehead. The three-eyed raven is played by a real-life raven, but its third eye is digitally added in post-production.
In the books
In the A Song of Ice and Fire novels, the bird is a three-eyed crow rather than a raven. The crow is more active in Bran's dreams; it helps him wake from his coma. The crow speaks to Bran, telling him it can teach him how to fly, other times it screeches the words "fly or die".