Sight/GoT

: Jojen Reed: "I have the Sight. I can see things."
: Karl Tanner: "That's a fine thing. A fine thing."
: Jojen Reed: "Things that haven't happened yet."
: — Jojen Reed explains Greensight to Karl Tanner[src]
"Greensight" (or just "the Sight") is the ability to perceive future, past, or contemporary but distant events in dreams. It was one of the talents of the greenseers of the Children of the Forest. Some humans are also born with this ability.
Bran Stark experienced a prophetic dream that his father was in the crypts of Winterfell when Eddard Stark was indeed executed in King's Landing. His younger brother Rickon also experienced a similar dream. Jojen Reed later explained to Bran that this was greensight, and that he had a similar dream the day Eddard was killed. Upon being told about this dream Howland Reed, his father, openly wept, realizing that it meant that his great friend Eddard had died.
== In the books ==
In the A Song of Ice and Fire novels, greensight seems to be limited to people who are blood of the First Men, i.e. the Free Folk and the Northmen (including the Crannogmen). This may have some connection with the fact that the First Men lived in Westeros for centuries alongside the magical Children of the Forest, and even adopted their religion of the Old Gods of the Forest, but the later Andals killed or drove away the last of the Children of the Forest during their invasion of the continent six thousand years ago.
Other peoples across the world, however, do claim to have their own various means of experiencing visions of the past, present, and future. The Warlocks of Qarth claim similar abilities - particularly their leadership known as the Undying. Red Priests of R'hllor, the Lord of Light - such as Melisandre - claim to receive ecstatic visions from communion with their flames. Members of the Targaryen bloodline were known to frequently experience prophetic dreams - in fact, it was due to a prophetic dream that the Targaryens survived the Doom in the first place. This trait was preserved among the Targaryens - along with, allegedly, the ability to bond with and ride dragons - due to "keeping the bloodline pure" through heavy inbreeding. Even generations after the Targaryen dragons died out, members of the bloodline would often have vivid "dragon dreams" in which they saw the beasts alive.
Thus many different peoples and diverse religions claim the power of prophecy, but the exact nature of each is deliberately vague within the narrative. It isn't clear if the prophetic dreams that Daenerys Targaryen experiences are simply "greensight" known by a different name, or a variant ability, or an essentially unrelated phenomenon. Whatever the case, the term "greensight" is usually only used by the descendants of the First Men such as the Northmen and wildlings.