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.