Ooh I know some of this!

Sauron was a Maia (lesser primordial spirit), who along with Morgoth (a fallen spirit formally known as Melkor) was at odds with essentially every other spirit and the general concepts of peace and tranquility that were sought by the creators of the universe. After Morgoth was defeated, Sauron inherited the role of the eponymous Dark Lord and sought to rule Middle-Earth in its entirety.

The Wizards were the Istar, a group of Maiar tasked by the Valar (greater spirits) and Manwe (the king of the Valar) to travel to Middle-Earth and aid the Free People in their fight against Sauron. They took the form of elderly men and roamed the lands to counter and subvert Saurons influence. Their mandate prevented them from open conflict, which is why they took on the role of advisors and supporters instead of just fighting Sauron head on or rallying armies to fight him. Their origins were unknown to any of the Free People, but the Elves for sure new that they weren't just Men - since they lived for thousands of years and had gifts that no mortal Men possessed.

Bombadil is likely another Valar, off in Middle Earth doing his own thing - in ancient ages many of the Valar visited or lived in Middle Earth, so it could be that he didn't return to Valinor and just hung around. His complete disinterest in intervening in the conflict is one clue, and the fact that he exceeds Sauron in raw power as the One Ring is completely mundane to him, whereas Gandalf fears that it will overpower him, is another.