Radagast the Brown was one of the wizards sent to Middle-earth in around T.A. 1000 in order to assist the free peoples of Middle-earth against Sauron.1
Radagast was a Maia known as Aiwendil.1
He was created by Ilúvatar before the Music of the Ainur and was among the Maiar who entered into Arda to help the Valar shape the world. Aiwendil was associated to the Vala Yavanna and shared her love for both plants and animals. This is reflected in his name Aiwendil which is Quenya for "lover of birds"
Not even after the Downfall of Númenor and the separation of the Undying Lands for Arda did the Valar seize to take council on the future of Middle-Earth.1
When the shadow of Sauron began to reappear it was decided that they would send emissaries to Middle-earth to assist the peoples of Middle-earth. For this order, known as the Istari, Aulë chose Curumo (Saruman), Oromë chose Alatar and Pallando, and Manwë chose Olórin (Gandalf). Yavanna subsequently persuaded Curumo to take Aiwendil with him.
The Istari first appeared in Middle-earth around T.A. 1000.1
Saruman arrived first and Radagast at the same time as Gandalf. The Istari were restricted to the bodies of old men and subject to the weaknesses of a physical body. Although immortal, their physical bodies could be destroyed by violence. They were forbidden to reveal their true power and were ordered to use their wisdom to persuade Men and Elves to do good.
According to Gandalf, Radagast never travelled much, and settled down in Rhosgobel on the western borders of Mirkwood, between Carrock and the Old Forest Road.1, 2
Radagast cared mostly for animals and birds and became friends with the great eagles and Beorn the Skin-changer.2, 3
In T.A. 2851, Gandalf revealed that the Necromancer was indeed Sauron to the White Council.4
He urged an immediate attack on Dol Guldur but was overruled by Saruman. After this, Saruman started searching the Gladden Fields for the One Ring. Believing Saruman's intentions were pure, Radagast decided to aid his search with beasts and birds.5
In T.A. 3018,4
Radagast met Gandalf not far from Bree and delivered a message from Saruman asking Gandalf to come to Orthanc. It does not appear that Radagast was aware of Saruman's plan for Gandalf. Indeed he later helped him escape by arranging for birds and beasts to send messages to Orthanc - Gandalf escaped upon the eagle Gwaihir.2
Whether or not Radagast played a further part in the War of the Ring is not known. The question if Radagast failed or not has been widely discussed.
1 - J.R.R. Tolkien, Christopher Tolkien (ed.), Unfinished Tales, "The Istari"
2 - J.R.R. Tolkien, The Lord of the Rings, The Fellowship of the Ring, "The Council of Elrond"
3 - J.R.R. Tolkien, The Hobbit, "Queer Lodgings"
4 - J.R.R. Tolkien, The Lord of the Rings, Appendix B, The Third Age
5 - J.R.R. Tolkien, Christopher Tolkien (ed.), The Silmarillion, Of the Rings of Power and the Third Age