Radagast the Brown was one of five Wizards sent to Middle-earth to contest the will of Sauron. Originally a Maia of Yavanna, Radagast mainly concerned himself with the well-being of the plant and animal worlds, and thus did not participate heavily in the War of the Ring.
After the Undying Lands were separated from Arda during the Downfall of Númenor, Manwë was still concerned for the fate of the peoples of Middle-earth. Even though Sauron had been overthrown, he had not been permanently vanquished. Given time, his shadow began to fall again. A council of the Valar was summoned and it was decided that they would send emissaries to Middle-earth. These messengers should be "mighty peers of Sauron, yet forgo might, and clothe themselves in flesh," as they were intended to help Men and Elves unite against Sauron. The wizards were forbidden from matching the Dark Lord in power and fear. Aulë chose Curumo (Saruman), Oromë chose Alatar and Pallando (the Blue Wizards), and Manwë chose Olórin (Gandalf). Yavanna subsequently begged Curumo to take Aiwendil with him.
Around the year TA 1000, the Maiar arrived at the Grey Havens in the west of Eriador, having the form of old Men, whom the peoples called Wizards. Curumo arrived first and alone, and Aiwendil arrived at the same time as Olórin. When the Wizards took their mission, they roamed Middle-earth. In this period, Aiwendil had been renamed "Radagast" by the Ñoldor and was known for both his protection of the great Forests and his zeal for the animals, worrying little with the affairs of Men and Elves, but was far more knowledgeable in plants, birds and beasts in the forest. He also turned away from Saruman during this period, unaware that the White Wizard despised him and considered him a fool.
In general, Radagast was never much of a traveler. There is not much to be told about his early journeys, but by the late Third Age he eventually settled down and dwelt, for a time at least, at Rhosgobel somewhere between the Carrock and the Old Forest Road. Situated on the western borders of Mirkwood, it can be assumed that the Wizard held watch against the Shadow of Dol Guldur that slowly engulfed the forest. It is likely that he became acquainted with the inhabitants of that region. Close to animals and birds, he was friends with the Great Eagles. Although the neighboring Beorn was unsociable, he used to see the wizard from time to time, and considered Radagast "not bad" for a Wizard.
In TA 2851 the White Council met once more and, after that, Saruman began to search the Gladden Fields for the One Ring. Radagast decided to aid his search with birds and beasts who acted as spies hoping that Saruman's actions would help watch and hinder Sauron. Radagast did this in good faith, knowing nothing of Saruman's real ambitions to keep the Ring for himself.
By the time of the War of the Ring Radagast did not dwell any more in Rhosgobel. In summer TA 3018 Saruman told Radagast that he was willing to help Gandalf, and sent the Brown Wizard to seek him out at once. Radagast did not know much of Eriador but sought for the Shire, knowing that he would find Gandalf nearby.
Indeed, on Midsummer's Day, Radagast was sitting on the side of the Greenway with his horse near Bree when Gandalf found him on his way to the village. Radagast warned Gandalf that the Nazgûl were abroad, disguised as riders in black, and that they were seeking news of the Shire. He also gave him Saruman's invitation and agreed to help Gandalf by getting beasts and birds to send news to Orthanc. With that he rode away back towards Mirkwood.
By sending Gandalf to Orthanc, Radagast unwittingly had him captured. Saruman's message proved to be a trap for Gandalf who was imprisoned in Orthanc, but still he did not believe that Radagast was also a part of Saruman's plans. Indeed, it was thanks to Radagast that Gandalf was able to escape from the pinnacle of Orthanc upon the wings of Gwaihir.
Radagast's actions during and after the rest of the War are not recorded. After the Council of Elrond] many scouts were sent out from Rivendell to many different locations. Some passed over the Misty Mountains and eventually came to Rhosgobel, but they found that Radagast was not there. His fate after the War of the Ring is not known.