It's not known what their exact position of power is in the Fairy World hierarchy, but all other fairies, especially Jorgen, all appear to be subjects to their whim. With that being said, considering the positions that they have, and the fact that basically every fairy in Fairy World are under their authority and guidance, the members of The Fairy Council all appear to be among some of the most influential and powerful fairies in all of Fairy World (and quite possibly some of the most powerful beings in the entire series.)
The four members of The Fairy Council all wear full-body robes that shroud their full faces and almost all of bodies (with the exceptions being their arms), and are each colored from one of the prominent colors in the Fairy World palette (turquoise, pink, blue, and light purple) with the exception of yellow. Another rather unique trait that they have is that unlike the other fairies that reside in Fairy World, All four of them are shown to be so powerful that they do not seem to have a need for wands and instead use their hands to control their magic. They also bear small golden crowns that rest just above their heads like all other fairies.
Very little is known about the history of The Fairy Council; however, in all references to them, they are always depicted as the true, absolute authority throughout all of Fairy World. Two known major events in history that demonstrate their roles are that they are the founders of the very first set of Rules that all fairies and godchildren must follow, and that they led the fairies to victory in the war against The Darkness.
In the episode, Fairly Odd Baby, it is revealed that they decreed that no more fairy babies would be born in Fairy World after the incident regarding Cosmo. However, Jorgen wrote this new rule on a sticky pad and forgot to officially put it down (making having a fairy baby legal.) They are also responsible for vanquishing The Darkness. They were first mentioned in Abra-Catastrophe!, but they were never officially revealed until the Wishology trilogy.
They also make another appearance in the special episode, Timmy's Secret Wish!, where they act as the judges of Timmy's trial.
Live-action versions of the Fairy Council, without their hoods, appeared in Grow Up, Timmy Turner! to reveal the Timmy Turner Loophole to Timmy and Tootie at the end of the movie. However, in the film, the turquoise hooded member of the council was absent (being replaced by Jorgen), and that the pink hooded member of the council was a female instead of a male (all of the members of the council are shown to be men.)
The episode, "Man's Worst Friend", proves that they either have control over the Anti-Fairy Council or at least have access to their chambers because they were the ones who ordered Foop to swap the pets in Dimmsdale. It is unknown why they did this or why the Anti-Fairy Council would take credit for this deal.