The Valar and Maiar were both Ainur.
"With the Valar came other spirits whose being also began before the World, of the same order as the Valar but of less degree... their servants and helpers." [Valaquenta]
Of the Valar it is said that they are "the Powers of the World" since
"their power should thenceforward be contained and bounded in the World, to be within it forever, until it is complete, so that they are its life and it is theirs." [Ainulindalë]
[The Maiar were called forth by Manwë and the other Valar,] "he called unto himself many spirits both greater and less, and they came down into the fields of Arda and aided Manwë, lest Melkor should hinder the fulfillment of their labour for ever... the Valar drew unto them many companions , some less, some well nigh as great as themselves, and they labored together in the ordering of the Earth and the curbing of its tumults." [Ainulindalë]
The Silmarillion states "the Valar drew unto them many companions, some less, some well nigh as great as themselves" [Ainulindalë] These spirits were the Maiar. Of the 13 Valar "Nine were of chief power and reverence... surpassing beyond compare all others, whether of the Valar and the Maiar, or of any other order that Ilúvatar has sent into Eä." [Valaquenta] These were Melkor, Manwë, Varda, Ulmo, Yavanna, Aulë, Mandos, Nienna, and Oromë. The Maiar are known as the companions of the Valar, but of "a less degree". Some Maiar may come close to matching the Valar, but in this case only those six Valar who were not included among the chiefs which would include Tulkas, Lórien, Estë, Vairë, Vána and Nessa.
I'd say that since none come close to the chief who are "surpassing beyond compare all others" then the reference to some of the Maiar being "well nigh as great as themselves" has to be in reference to the lesser Valar. Therefore there would be Maiar who come close in comparison to the Valar, but only the lesser Valar [Tulkas, Lórien, Estë, Vairë, Vána and Nessa], but so far as I can tell none are mightier than any Vala.
The Silmarillion says, "ye shall show forth your powers in adorning this theme, each with his own thoughts and devices, if he wills" [Ainulindalë] It is also said that they are called the Valar, "the Powers of the World", since "their power should thenceforward be contained and bounded in the World, to be within it forever, until it is complete, so that they are its life and it is theirs." [Ainulindalë] We have seen Aulë create Dwarves but they could only be as automatons as far as he could go with them. The Valar generally communicate with each other through telepathy, but when they put on a physical raiment this mode of communication does become more difficult in practice.
The Ainur were the Powers great and small who worked "to fulfilled the vision which they had seen" [ValaQuenta] which was Ea. The Maiar were "spirits whose being also began before the World, of the same order as the Valar but of less degree" [Valaquenta]. With the Valar, to whom they are "servants and helpers" [Valaquenta] they saw the vision of Earth and entered in it to bring order.
Maiar clad themselves in raiment similar to the Children of Illuvatar or "they may clothe themselves in their own thought, made visible in forms of majesty and dread" [Ainulindalë] and its said that "the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present." [Ainulindalë]
Like the Elves they are bound to Arda.
But this condition Illuvatar made, or it is the necessity of their
love, that their power should thenceforward be contained and bounded
in the World, to be within it for ever, until it is complete, so that
they are its life and it is theirs. [Ainulindalë]
The Istari were sent to rally the free peoples of the world to fight together. They were:
bidden to advise and persuade Men and Elves to good, and to seek to
unite in love and understanding all those whom Sauron... would
endeavor to dominate and corrupt. [UT, The Istari]
They arrived in the form of old men although they were full of vitality.
They came therefore in the shape of Men, though they were never young
and aged only slowly, and they had many powers of mind and hand. [UT,
for long they went about in simple guise, as it were of Men already
old in years but hale in body [The Istari]
and, "Men perceived that they did not die, but remained the same" although they "aged only by the cares and labours of many long years" [The Istari].
They are Maiar who wear the shapes of old men so as not to come in a display of power. They were to rally the peoples against the enemy but not themselves go head to head with him. See Appendix B in LotR. Gandalf did confront the Balrog of Moria and unleash his powers there. This is an example of two Maiar fighting one another.
Gandalf was originally the Maia Olorin. It is said that,
Wisest of the Maiar was Olorin… his ways took him often to the house
of Nienna, and of her he learned pity and patience… he loved the
Elves, he walked among them unseen, or in form as one of them, and
they did not know whence came the fair visions or the promptings of
wisdom that he put into their hearts. In later days he was the friend
of all the Children of Illuvatar, and took pity on their sorrows [Sil,
In the downfall of Sauron and evil, "these things were achieved for the most part by the counsel and vigilance of Mithrandir" [Of the Rings of Power and the Third Age]. He had some of the same qualities as Gandalf and Olorin, counseling the free people with his wisdom and having pity on them. As Gandalf he, "was closest in friendship with the Eldar" just like he used to be when he walked among them as Olorin.
Sauron himself was a Maia of Aule and the Istari were Sauron's peers but on a lesser level. Saruman himself was also a Maia of Aule so Sauron could figure him out. There is also Melian from whom Galdriel learned many things. She had the famous Girdle of Melian up to protect Doriath which was the Elf kingdom that she and Thingol ruled. Orome who was a vassal of Ulmo guided the Elves to Aman when he came upon them in his travels.