The Istari are sent with a mission on Arda, Gandalf is sent to help humans, elves, dwarfs and all other races to vanquish Sauron. What are the roles of Saruman, Radagast and maybe the other two blue wizards (Alatar and Pallando).
Since Saruman is corrupted by the power of the ring, and Radagast is guarding Greenwood, what are their other roles? Their higher calling.
ALL the Istari were there to resist Sauron; Gandalf was the only one who remained true to his mission.
In a pieced-together essay in Unfinished Tales called "The Istari", Tolkien said:
Indeed, of all the Istari, one only remained faithful, and he was the last-comer. For Radagast, the fourth, became enamoured of the many beasts and birds that dwelt in Middle-Earth, and forsook Elves and Men, and spent his days among the wild creatures...And Curunír'Lân, Saruman the White, fell from his high errand, and becoming proud and impatient and enamoured of power sought to have his own will by force, and to oust Sauron; but he was ensnared by that dark spirit, mightier than he.
But the last-comer was named among the Elves Mithrandir, the Grey Pilgrim, for he dwelt in no place, and gathered to himself neither wealth nor followers, but ever went to and fro in the Westlands from Gondor to Angmar, and from Lindon to Lorien, befriending all folk in times of need.
The remaining two, the Blue Wizards who we only know by their names in Valinor as Alatar and Pallando, are said to have gone into the east; but what they did there, we do not know. Tolkien said in a letter that they may have inspired "secret cults and "magic" traditions that outlasted the fall of Sauron."
It is interesting to wonder what might have happened had Saruman and Radagast remained true to their mission. Could all three Istari have travelled with the Fellowship? If they had, could they together have defeated the Balrog more easily, and been with Frodo when he had to make his final choice? With three Istari involved, might Gondor and Rohan have faced the forces of Mordor with greater strength? We will never know.
According to The Silmarillion, helping the resistance to Sauron was the mission of all the Istari.
Even as the first shadows were felt in Mirkwood there appeared in the west of Middle-earth the Istari, whom Men called the Wizards. None knew at that time whence they were, save Círdan of the Havens, and only to Elrond and to Galadriel did he reveal that they came over the Sea. But afterwards it was said among the Elves that they were messengers sent by the Lords of the West to contest the power of Sauron, if he should arise again, and to move Elves and Men and all living things of good will to valiant deeds.
The Silmarillion: Of the Rings OF Power and the Third Age
There is a similar description of their role in the introduction to the section of Appendix B that deals with the Third Age:
When maybe a thousand years had passed, 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 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.
The Lord of the Rings Book 6, Appendix B: The Tale of Years
Both quotes describe the Istari as sharing a single mission: to help those who opposed Sauron.
The other answers address Tolkien's thoughts shortly after finishing The Lord of the Rings, when he decided to highlight Gandalf's success by making him the only of the Istari to succeed on his mission. Later Tolkien rethought this and in the final year of his life he sketched out a new history for the Blue Wizards.
In Tolkien's final conception, the Istari were sent in two groups.
The first group was dispatched in the Second Age, and consisted of the two Blue Wizards. They were sent to the East to help guide the tribes of men who were rebelling against Morgoth's rule, and thus prevent Sauron from getting a firm hold on the region.
The 'other two' came much earlier, at the same time probably as Glorfindel, when matters became very dangerous in the Second Age. Glorfindel was sent to aid Elrond and was (though not yet said) pre-eminent in the war in Eriador. But the other two Istari were sent for a different purpose. Morinehtar and Romestamo. Darkness-slayer and East-helper. Their task was to circumvent Sauron: to bring help to the few tribes of Men that had rebelled from Melkor-worship, to stir up rebellion ... and after his first fall to search out his hiding (in which they failed) and to cause [?dissension and disarray] among the dark East ... They must have had very great influence on the history of the Second Age and Third Age in weakening and disarraying the forces of East ... who would both in the Second Age and Third Age otherwise have ... outnumbered the West.
The Peoples of Middle-earth - "The Five Wizards"
The reason why the Blue Wizards were sent in the Second Age, was that Sauron was then trying to adopt a guise of a good person, and thus wasn't able to go to the East to recruit himself. Morgoth hadn't been there since the early first age, and so now was the ideal time to try causing a disruption there before Sauron could come back and fix things. So by beginning their mission then, the Blue Wizards were able to more effectively repel Sauron from gathering massive armies there.
But until [S.A.] 1600 he [Sauron] was still using the disguise of beneficent friend, and often journeyed at will in Eriador with few attendants, and so could not risk any rumour that he was gathering armies. At this time he perforce neglected the East (where Morgoth’s ancient power had been) and though his emissaries were busy among the multiplying tribes of eastern Men, he dared not permit any of them to come within sight of the Númenóreans, or of Western Men.
The Orcs of various kind (creatures of Morgoth) were to prove the most numerous and terrible of his soldiers and servants; but great hosts of them had been destroyed in the war against Morgoth, and in the destruction of Beleriand. Some remnant had escaped to hidings in the northern parts of the Misty Mountains and the Grey Mountains, and were now multiplying again. But further East there were more and stronger kinds, descendants of Morgoth’s kingship, but long masterless during his occupation of Thangorodrim, they were yet wild and ungovernable, preying upon one another and upon Men (whether good or evil). But not until Mordor and the Barad-dûr were ready could he allow them to come out of hiding, while the Eastern Orcs, who had not experienced the power and terror of the Eldar, or the valour of the Edain, were not subservient to Sauron – while he was obliged for the cozening of Western Men and Elves to wear as fair a form and countenance as he could, they despised him and laughed at him. Thus it was that though, as soon as his disguise was pierced and he was recognized as an enemy, he exerted all his time and strength to gathering and training armies, it took some ninety years before he felt ready to open war. And he misjudged this, as we see in his final defeat, when the great host of Minastir from Númenor landed in Middle-earth. His gathering of armies had not been unopposed, and his success had been much less than his hope. But this is a matter spoken of in notes on “The Five Wizards”. He had powerful enemies behind his back, the East, and in the Southern lands to which he had not yet given sufficient thought.
The Nature of Middle-earth - "Note on the Delay of Gil-galad and the Númenóreans"
The other three wizards, Radagast, Saruman, and Gandalf, were sent in the third age, after Sauron had already become a threat. Their role can be seen directly in the book.
The first group was a preventive measure that was done to limit Sauron's ability to grow before he even came in power. The second group was to contest him after he was already around.
