It was a continent that lay to the west of Middle-earth across the great ocean Belegaer, although it lay in another dimension during the time of The Lord of the Rings. It is the home of the Valar, and three kindreds of Elves: the Vanyar, some of the Noldor, and some of the Teleri. The island of Tol Eressëa lies just off the eastern shore.
Upon the destruction of Almaren in very ancient times, the Valar fled to Aman, and there established the realm of Valinor. Seeking to isolate themselves, they raised a great mountain fence, called the Pelóri, on the eastern coast, and set the Enchanted Isles in the ocean to prevent travellers by sea from reaching Aman.
For reasons unknown, the Valar left two lands outside the wall of the Pelóri: Araman to the northeast and Avathar to the southeast. Ungoliant, an ancient being in the form of a great spider, had managed to escape notice in Avathar. When Melkor was released from his captivity, he fled to Avathar, scaled the mountains with the help of Ungoliant, and wrought destruction of great consequence in Aman. Melkor, or Morgoth as he is also called, persuaded Ungoliant to kill the Two Trees of Valinor and take from them what energy she could to quench her hunger, as Ungoliant (see also Shelob) was always hungry.
The first mortal man to succeed in navigating to and passing the Isles of Enchantment was Eärendil, who came to Valinor to seek the aid of the Valar against Melkor, now called Morgoth. His quest was successful, the Valar went to war again, and also decided to remove the Isles.
Soon after this, the great island of Númenor was raised out of Belegaer, close to the shores of Aman, and the Three Houses of the Edain were brought to live there. Henceforth, they were called the Dúnedain, or Men of the West, and were blessed with many gifts by the Valar and the Elves of Tol Eressëa. The Valar feared — rightly — that the Númenóreans would seek to enter Aman to gain immortality (even though a mortal in Aman remains mortal, due to it not being their final destination), so they forbade them from sailing west of sight of the westernmost promontory of Númenor. In time, and not without some corrupting help from Sauron, the Númenóreans violated the Ban of the Valar, and sailed to Aman with a great army under the command of Ar-Pharazôn the Golden. Eru collapsed a part of the Pelóri on this army, trapping it but not killing it. It is said that the army still lives underneath the pile of rock.
In light of this new development, Eru removed Aman from the spheres of the world. The earth, at this time, was flat. They split it in two, and made the half containing Middle-earth round, so that a mariner sailing west along Eärendil's route would simply emerge in the far east. For the Elves, however, they crafted a Straight Road that peels away from the curvature of the earth and passes to the now-alien land of Aman. A very few non-Elves are known to have passed along this road, including Frodo Baggins, Bilbo Baggins, Samwise Gamgee, and Gimli.