Greensight is the ability to have prophetic dreams called green dreams.
A person with greensight sometimes dreams as other people, but the green dreams are different, filled with symbolic meaning, images, and metaphors of what is to come.
The meaning behind the dreams is not always obvious, but the dreamer experiences the fulfillment of visions in the unfolding of events. Supposedly these dreams can concern the dreamer or another person, but the dreamer will be able to tell the difference. Greenseers might also dream of their own deaths. Wargs have been known to also possess this ability.
Greenseers, the wise men of the children of the forest, reputedly possessed the greensight. Some crannogmen are also known to have greensight. One of them is Jojen Reed, who has unnaturally green eyes. It is possible that this is the result of his advanced greensight abilities.
My brother has the greensight. He dreams things that haven't happened, but sometimes they do.