The Eagle is a nameless character in Spirit: Stallion of the Cimarron. The eagle is affiliated with Spirit's Herd, particularly Spirit himself. The eagle is shown to be very attached to Spirit, and searches tirelessly for him when he is taken from the herd.
Without him, we would never understand how fast and wild Spirit really is. He is a major character because it shows how before white invaders came and wrecked all of the land that so obviously belonged to them, the animals lived together in peace.
He seems to care for Spirit, but he was obviously hurt when he saw Spirit in a new place with horses. He must have felt that Spirit left him and his herd behind, so he turned back. He was very happy when Spirit came back.