The Ainulindalë is the title of the first part of The Silmarillion.
The supreme deity of Tolkien's universe is called Eru Ilúvatar (The One who is Father of All). The tale began with Ilúvatar's creation of spirits of lesser power than him, yet of independent nature, named the Ainur (The Holy Ones). Ilúvatar made divine music with them ("Ainulindalë" translates from Quenya as the Music of the Ainur). Melkor, who was then one of the Ainur, broke the harmony. Then Ilúvatar began the music once more, in which Manwë Sulimo, Melkor's brother, sang the leading part. Yet again, the harmony was broken.
Finally, Ilúvatar began a third theme which the Ainur could not comprehend since they were not the source of it. Ilúvatar then ceased the music, and showed to the Ainur the essence of what their song symbolized, the history of a whole world. This is known as the vision of Ilúvatar. The Ainur became fascinated by it, and asked Ilúvatar to put it into being.
Ilúvatar muttered "Eä", that is, "be!". This order created the universe, and therefore the universe became known as "Eä". Four of the Ainur who possessed the greatest power formed a region of Eä which they called Arda, a world which was to be the scene of all the following tales of the Silmarillion. Melkor, who was one of the four, attempted to take it for his own. However Manwë, who was also one of the four asked for the help of other Ainur. They descended to Arda too and helped him and thus Melkor was forced to flee to some unknown part of Eä.
To the mortal Men Eru gave the gift of freedom. So when they die their souls go somewhere beyond the known existence, perhaps to sing along with Eru Ilúvatar the second Music of the Ainur; this is the reason why Men are not constrained by the fate of the Music like the Elves and other beings of Arda are.