I just restarted the Silmarillion today on my way home from work. Shaw's voice is so right for this piece. Its magical.
Anyway, Steak asked for the hierarchy, so I paid close attention for him.
The Ainur encompasses the Valar, the Maiar and the other creatures of Iluvatar's creation before the Ainulindule and the creation of Arda. When Iluvatar gave the Ainulindule to the Ainur, he gave each a part and bade them sing it.
I had been under the misimpression that Arda was created from the Ainulindule, but that is not the case. From the Ainulindule, Iluvatar wove a vision of what could be for the Ainur, then created Arda, the Earth and told the Ainur to go make it as they had seen in the vision.
Fifteen Ainur went to Arda and became part of it and it a part of them. Of the fifteen one was cast out and the other fourteen are named the Valar. There are seven male and seven female Valar.
The Valar (kings) were Manwe, Ulmo, Aule, Orome, Mandos, Lorien and Tulkas. The Valiye (queens) were Varda, Yavana, Niena, Este, Vaire, Vana, and Nessa. Of the Valar, eight were called the 'High Ones': Manwe, Varda, Ulmo, Yavanna, Aule, Mandos, Niena and Orome.
Interestingly, four of the Ainur that came to Middle Earth are directly connected to elements: Manwe is associated with the air, Aule with the earth, Ulmo with the water, and Melkor with fire.
During the creation of Arda, Melkor and Aule fought constantly. Aule would create an ocean and Melkor would drain it. Aule would build mountains and Melkor would knock them down. Aule would dig a great canyon and Melkor would fill it.
As for the Maiar, it turns out that all the Valar--and Melkor--brought some with them. Sauron was one of Aule's earth Maia who was seduced away by Melkor (now called Morgoth).