Radagast, like the other Wizards, came from Valinor around the year 1000 of the Third Age of Middle-earth and was originally one of the Maiar. His name as a Maia was Aiwendel, meaning "bird-friend" in Quenya. The Vala Yavanna forced Saruman to accept Radagast as a companion, which may have been one of the reasons Saruman contempted him.
According to the essay "The Istari" from the Unfinished Tales, the name Radagast means "tender of beasts" in Adûnaic, the language of Númenor. However, in a later note Tolkien said that the name is in the language of the Men of the Vales of the Anduin, and that its meaning is not interpretable.
Radagast was unwittingly used by Saruman to lure Gandalf to Orthanc, where Gandalf was captured; however Radagast also unwittingly helped rescue the grey wizard by alerting the Eagles of Gandalf's journey there.
The name Radagast may also be Anglo-Saxon. The word gast means "ghost, spirit, angel." The element rad could be derived from rudu, meaning "ruddy, reddish." If this is the case, his name could be translated as 'Ruddy Angel'. Since the Maiar are Tolkien's 'lesser angels', Radagast would mean simply 'Ruddy Maia', perhaps in reference to his cloak's color (Brown, or perhaps ruddy brown). But this may be too much speculation. See Radagast (god) for meaning of his name in Slavic languages from which Tolkien possibly drew the name.
It is not known when or if he left Middle-earth. Tolkien writes that he forsook his mission as one of the Wizards by becoming too obsessed with animals, so presumably he failed, and would not be allowed to return with honour like Gandalf did.