Radagast: Radagast the Brown is one of the Istari (called Wizards by Men) that were sent to Middle-earth by the Valar to fight Sauron in the third age. Radagast was a peer of Gandalf the Gray, Saruman the White, and two other Istari who went into the east and of whom little is known (Alatar and Pallando, the Blue Wizards).
To understand what Radagast is we need to look at the higher beings of Middle-earth (discussed primarily in The Silmarillion). Iluvatar is "the father of all," the creator of all else in the world. Iluvatar's first creations were the Ainur, who are powerful spirits that helped create Arda, the world that contains Middle-earth. Some of the Ainur lived in and shaped Arda after its creation; these were classified as two types: Valar (Vala singular) were the "greater," more powerful Ainur, and Maiar (Maia singular) were lesser Ainur. Radagast and the other Istari were all Maiar. Radagast was a Maia under Yavanna, the Vala most associated with the animals and plants of Middle-earth.
The Istari were intended by the Valar to aid the people of Middle-earth in their fight against Sauron, and they appeared as "old but vigorous" men. Gandalf (Mithrandir) is probably the best known of the Istari, as he directly helped both Elves and Men destroy the One Ring. Radagast, on the other hand, played little known role in the War of the Rings. A good summary of the Istari is found in Appendix B of The Lord of the Rings,
"When maybe a thousand years had passed [in the third age], and the first shadow had fallen on Greenwood the Great, the Istari or Wizards appeared in Middle-earth. It was afterwards said that they came out of the Far West [Valinor] and were messengers sent to contest the power of Sauron and to unite all those who had the will to resist him; but they were forbidden to match his power with power, or to seek to dominate Elves or Men by force and fear.Radagast's sole appearance in The Lord of the Rings is in book II, where an encounter with him is described by Gandalf at the Council of Elrond. Radagast was asked by Saruman to find Gandalf and tell him that he was needed at Isengard, Saruman's home, immediately. Radagast met Gandalf near Bree and told him that the Nine Riders (the Nazgul) were traveling abroad again, searching for the Shire, and that Gandalf should quickly go to Isengard. Radagast left immediately (quite clearly afraid of the Nine Riders), but before he left Gandalf asked him to have all birds and beasts send news to Isengard. It was through Radagast's fulfillment of this request that Gwaihir the Great Eagle came bearing news and rescued Gandalf from Saruman's imprisonment on top of Orthanc. Radagast was later sought by the Elves during the War of the Rings, but could not be found.
"They came therefore in the shape of Men, though they were never young and aged only slowly, and they had many powers of mind and hand."
Although Saruman had by that point in time turned to evil, Radagast was apparently still honest and true; Saruman deceived Radagast into thinking that Saruman's intentions regarding Gandalf were good. As Gandalf later says, "It would have been useless in any case to try and win over the honest Radagast to treachery."
While Gandalf spent most of his time with the Elves, and Saruman with Men, Radagast cared most for the animals and plants of Middle-earth. Gandalf describes Radagast as "a worthy wizard, a master of shapes and changes of hue; and he has much lore of herbs and beasts, and birds are especially his friends." The Silmarillion calls Radagast "the friend of all beasts and birds." He's considered to have been generally useless in the fight against Sauron (e.g. he knows the Nazgul are about, yet does nothing to help Gandalf but send news). It seems likely that he spent the War of the Rings among the animals and plants of Middle-earth, protecting them from Sauron's creatures, though I know of no records saying such. If he had lived in our time he probably would have been a biologist.
Radagast is completely absent from Peter Jackson's Lord of the Rings movies. In the movies Gandalf goes to seek counsel from Saruman without being summoned, and escapes from the top of Orthanc by catching, and speaking with, a moth who then returns with an eagle.
Rhosgobel: Rhosgobel was Radagast's home in Middle-earth at one time. The Lord of the Rings contains little information on the location of Rhosgobel other than that "it was near the borders of Mirkwood."
The Atlas of Middle-earth (a great book for anyone looking for maps of Middle-earth) shows Rhosgobel as being on the western borders of Mirkwood, north of the Old Forest Road. Mirkwood is east of the Misty Mountains and west of the Lonely Mountain, and Rhosgobel is near the Eagle's Eyrie found on the eastern side of the Misty Mountains. According to the map on page 80 of the Atlas of Middle-earth, Rhosgobel is about 275 miles north of Caras Galadon in Lorien, about 350 miles north of Fangorn Forest's northern border, about 175 miles east of Rivendell, and about 400 miles east of Weathertop (Amon Sul).
Other sites: While I absolutely love Tolkien's work, I am by no means an expert on Radagst, Rhosgobel, or anything from Tolkien's world, so here are a few sites with more information: