The Ainur are the Holy Ones, spirits created by Eru, called the Valar and Maiar.
Each had his or her own part and sang solo or in pairs, until Eru had them sing in harmony to create a vision of the world of Arda, also called Earth, not showing them the ending. The vision so enchanted them that many, including the most powerful, entered the formlessness and tried to make it material, each taking his or her part of the song of Arda, learning to harmonize their efforts partly due to the discord of Melkor. It will become perfect only after the Last Battle, when all the singers are present and understand the vision at last.
The most powerful members of the Ainur are called the Valar, and meet together in the Council of Mahanaxar. Of these, the greatest are called the Aratar, numbering nine until the fall of Melkor. These High Ones of Arda are Manwë and Varda, Ulmo, Yavanna and Aulë, Mandos, Nienna, and Oromë. In majesty they are peers, beyond all others. The Lords of the Valar are seven, in order: Manwë, Ulmo, Aulë, Oromë, Mandos/Namo, Lórien/Irmo, and Tulkas. The Queens of the Valar are also seven, and in order: Varda, Yavanna, Nienna, Estë, Vairë, Vána, and Nessa.
The Maiar are less powerful Ainur who aligned themselves with whichever Vala most suited their interests, sometimes more than one or switching. A Maia aids the chosen Vala in that portion of Arda of which he or she sang into being and maintains. Thus Ossë, whose interest is in the sea waters near shore and in ocean storms, aids Ulmo, who is the Vala of all waters including the sea. On the other side, the Balrogs aid Melkor whose might attracted them, staying with him as his ethics slowly changed and twisted theirs as well.
In earlier works and notes, the Valar were capable of having children who might be considered Valar as well. For instance Eönwë and Ilmarë were earlier called the son and daughter of Manwë and Varda. In the Silmarillion, the Valar had no children and these two were considerd Maiar instead. See the Maiar page for further information.
Ainur is plural; Ainu is singular.
The word "Valar" is plural and may refer to both male and female. When used in showing gender, the plural "Valar" is the masculine form and "Valier" is the feminine form. "Vala" and "Valie" are the singular forms.