In the books
Radagast is described as a nature-lover who's mainly concerned with the well-being of the olvar and kelvar (flora and fauna) of Middle-Earth, and thus he doesn't participate heavily in the War of the Ring. This is probably due to him being a servant of the goddess of olvar and kelvar. It's implied that he's specifically tasked to protect the fauna and flora even after Sauron's defeat. He lives in seclusion in Rhosgobel of Mirkwood.
Radagast doesn't appear in Tolkien's books, but is mentioned several times. He's described to be wiser than any Man in all things concerning herbs and beasts, able to speak the many tongues of birds, and a "master of shapes and changes of hue". According to Gandalf, he's "never a traveller, unless driven by great need", "a worthy Wizard", and "honest". For Gandalf he is an ally and confidant, but Saruman is very contemptuous of him.
In The Fellowship of the Ring, Radagast is unwittingly used by Saruman to lure Gandalf to Orthanc, only to be captured. Fortunately Radagast also helps in rescuing Gandalf by sending Gwaihir the Eagle to Orthanc with news of the movements of Sauron's forces. When Gwaihir sees Gandalf was imprisoned on the top of the tower, he carries him to safety. Later Radagast is intended to be summoned to attend the Council of Elrond, but is not found. Tolkien leaves his fate ambiguous.
Peter Jackson's film adaptation
While Radagast is not even mentioned in Peter Jackson's The Lord of the Rings film trilogy, he appears in The Hobbit film trilogy as a secondary character. He's portrayed by Sylvester McCoy and is described as a very eccentric yet powerful wizard who prefers the company of animals to those of others, at one point using his powers to heal a dying hedgehog. He's shown to be able to communicate with birds, allowing some to nest in his hair. He usually travels in a sled pulled by enormous rabbits.
In The Hobbit: An Unexpected Journey, Radagast witnesses the effects of the evil corruption spreading throughout Mirkwood and tracks it back to Dol Guldur. He discovers that a Necromancer (Sauron) has taken residence in the ruined fortress. He's attacked by the spirit of the Witch-king of Angmar, but he escapes with the Morgul blade taken from the Witch-king. He finds Gandalf and the Company of Thorin Oakenshield, informing the Grey Wizard of his discoveries and giving him the Morgul blade. Then he distracts with his sled the Warg riders hot on the Company's trail, allowing the others to escape.
In The Hobbit: The Desolation of Smaug, Radagast is summoned by Gandalf to inspect the tombs of the Nazgûl, only to find them empty. They travel to Dol Guldur where Gandalf sends Radagast to inform Galadriel of their discoveries before he enters the fortress.
In The Hobbit: The Battle of the Five Armies, Radagast arrives in Dol Guldur with the White Council to rescue Gandalf. While the others battle Sauron and the Nazgûl, he takes the wounded Gandalf to his house. Gandalf decides to go to Erebor and tells Radagast to gather the birds and beasts and Radagast gives him his Wizard staff and sets off on his journey. Radagast is last seen riding one of the eagles that arrive to the Battle of the Five Armies to help defeat the Orc army.