Eru is the supreme deity of Arda and Middle-earth. He is the single creator, above the Valar, but has delegated most direct action within Eä to the Ainur, including the shaping of the Earth (Arda) itself.
Eru is transcendent, and completely outside of and beyond the world. He first created a group of angelic beings, called in Elvish the "Ainur", and these holy spirits were co-actors in the creation of Arda through a holy music and chanting called the "Music of the Ainur".
Eru is an important part of the stories of The Silmarillion but is not mentioned by name in Tolkien's most famous works, The Hobbit and The Lord of the Rings (he is alluded to as "the One" in the part of Lotr's Appendix A that speaks of the downfall of Númenor).
Eru as the Creator God
Eru alone can create independent life or reality by giving it the Flame Imperishable. All beings not created directly by Eru, (e.g. Dwarves, Ents, Eagles), still need to be accepted by Eru to become more than mere puppets of their creator.
Elves and Men were created by Eru directly, without delegation to the Ainur, and they are therefore called "Children of Ilúvatar" (Eruhini). The Dwarves were created by Aulë with the belated approval of Ilúvatar, though they would be second to the Children of Ilúvatar. Ents were also created by Eru under the request of Yavanna who reasoned that Dwarves would want to fell trees. Melkor instilled some semblance of free will into his mockeries of Eru Ilúvatar's creations (Orcs and Trolls).
The first things that Ilúvatar created were the Ainur (from his thoughts). He then bade the Ainur to sing to him; this was called the Ainulindalë, or the music of the Ainur. Melkor created a discord to the music, and Iluvatar added three themes to the Ainulindale, which talked about the fate of Arda.
The next thing he made was Eä, the "World and All That Is", and Eä was in the Void (Melkor was drawn to the Void, and so at the beginning of Arda, he lay in the Void, or the Outside as it was sometimes called). He then gave the Ainur the option to go into Ea and fashion it as they will. The most powerful Ainur who chose to do so were called the Valar, and they controlled the shaping of the Arda. They could not make life, however, as Aulë proves, who was able to give only shape to the dwarves, while consciousness was given to them by Eru. The Valar were accompanied by the Maiar. The elves and men, however, came directly from Ilúvatar's thoughts, and are referred to in The Silmarillion as The First Children of Ilúvatar and the Second Children of Ilúvatar, respectively, while the dwarves would be the Adopted Children.
Interventions
Eru Iluvatar's illustration.
In the First Age, Eru created and awoke Elves as well as Men.
In the Second Age Eru, called by the Valar, buried King Ar-Pharazôn and his Army when they landed at Aman in S.A. 3319. He caused the Earth to take a round shape, drowned Númenor, and caused the Undying Lands to be taken "outside the spheres of the Earth". When Gandalf died in the fight with the Balrog in The Fellowship of the Ring, it was beyond the power of the Valar to resurrect him; Ilúvatar himself intervened to send Gandalf back.
In a letter written by Tolkien, he stated that Eru again intervened, this time in the Third Age, causing Gollum to trip and fall into the fires of Mount Doom while still holding the One Ring, thus destroying it.
Etymology
Eru meant "The One" or "Alone", and the epithet Ilúvatar meant "Father of All" in the Quenya tongue.
