( Writings about Radagast are only concerned with the Wizard Radagast, not the Maia known as Aiwendil in Valinor. )
Radagast settled, after arriving to Middle-Earth as the fourth of the Wizards, at Rhosgobel. He was mainly interested (and maybe his mission was) in the care of birds, beast, and plants. At least he spent his days among the wild creatures. His name is said to be a name derived / given from the men of the vales of Anduin, the ancestors of the Rohirrim. The name Radagast means, in the tongue of Númenorë, Tender of Beasts. His "nick- name" the Brown refers simply to his clothing that were earthen brown in color. His name indicates that the Men in the upper vales of Anduin once were under some influence of Númenorë. That would not be unlikely judging from the history of those people.
Radagast was chosen by Yavanna for his task in Middle-Earth. He did not contribute much, it seems, to the overthrow of Sauron but it might be worth remembering that his task and love was for beast, birds, and earth and in that task he was not unsuccessful. He was sent to Middle-Earth together with Curumo / Saruman but left him right after their arrival to Middle-Earth, maybe because he had a feeling that his and Saruman's will did not match very well. Saruman never did think highly of him. Radagast was the only wizard besides Gandalf to remained faithful to his task.
He arrived at Middle-Earth around Third Age 1000. What happened to him after the fall of Sauron is not specified, but it might be assumed that once the task was ended he returned to his home in Valinor as Gandalf did.
Related links.:Misc.: The Istari. Encyc.: Radagast.