After the birth of Aang, he became the young Avatar's guardian. The two were very close; he was described by Aang to be "kind of like his father", having been given a necklace as well from his young pupil.
At the ceremony where Aang was to accept his role as the Avatar, Gyatso was present, looking up to give Aang an extra vote of confidence. Despite this, the young Avatar ran away.
Some time after, Gyatso was killed in the Air Nomad Genocide, and Aang discovered his remains scattered in a field, which had a profound impact on him.