The Ainur ('Holy Ones,' singular Ainu), were the first beings created by Ilúvatar, the 'order' of the Valar and Maiar, made before Eä. Though spirits they have the power to take on Visible and tangible shapes when it suits their needs; when in physical form they can even face death, though at Ilúvatar's will either die or be brought back to life (compare Gandalf in Moria versus Saruman at Bag End).
Maiar: Also other Ainur followed the Valar into Arda, these were known as the Maiar. Though lesser in wisdom and power they were no lesser in love of Ilúvatar's music, through which Arda came to existence, nor to the living beings that were brought forth.
Many Maiar took on physical shapes. Most known are perhaps the Wizards: Saruman, Gandalf, Radagast, etc. However, more of the Maiar came to Arda and also to Middle-earth; Melian even wedded an Elf, Thingol, and thus added power to their descendants.