What Were Their Purposes?

[Boromir88]

The Wizards (or Istari) were sent to Middle-earth by the Valar to aid those who fought against Sauron, and to to defeat him. Though there were some conditions put on them:

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.~Appendix B: Tale of the Years; The Third Age

Of the 5 Istari sent, Gandalf was the only one to complete the task.

Saruman had became corrupt, broke the rules, his goal became to dominate over the 'pathetic' Elves and Men, just as Sauron wanted to.

Radagast, although was a good wizard and friend of Gandalf, he pretty much said forget about the reason he was there and he took a liking to herbs, plants, and beasts. Eventhough if Radagast does help out Gandalf, he's hands off as far as the mission of the Istari, which were to unite those who were against Sauron.

The Blue Wizards (Alatar and Pallando) not much is said about them, just that they went into the East and had started up cults. Later in Tolkien writes a little more saying they were influential in keeping down the numbers of Sauron's forces, as they were persuading Men in the East to not join with him. However, it still appears that only Gandalf succeeded in the task of the wizards.

I hope that helps a bit and welcome to the downs.