Arda consisted of land masses, chiefly the continents of Middle-earth and Aman, oceans and seas, and an atmosphere. It was a part of Eä, the created World (i.e., of the whole Tolkienian star system). Arda was created, together with the rest of Eä, through the Music of the Ainur, and was set apart as a dwelling place for the Children of Ilúvatar (that is, Elves and Men). When Eä was created, many of the Ainur chose to dwell within it as long as it should last, and they built Arda and chose to reside there. The most powerful of these Ainur were called the Valar, the Powers of the World.
Originally, Arda was flat, rather than spherical (though it still had mountains and valleys), and the Valar built it to a symmetric plan. They lit up Arda with great lamps, and the explosion of biological life that ensued became known as the First Spring. This period ended when Melkor launched a surprise attack, destroying not only the lamps but the layout of the world.
After this time, Arda was surrounded by a mighty ocean, Ekkaia or the Encircling Sea, and the continents were separated from each other by Belegaer, called the Great Sea. During the Years of the Trees, Valinor, the realm of the Valar occupying most of the continent of Aman, was lit up by the Two Trees. Middle-earth was dark. At first it was largely controlled by Melkor and his servants, but when the Elves awoke the Valar went to war against Melkor, defeating him and imprisoning him in the Halls of Mandos for "three ages". During this war, many of the westernmost areas of Middle-earth were drowned beneath the waves.
During the captivity of Melkor, the Valar invited the Elves to dwell with them in Aman. Many Elves accepted this invitation, and set out for the West; these became known as the Eldar. Along the way, some groups of Eldar, notably the Nandor and the Sindar, chose to remain in Middle-earth. The Sindar settled in the region of Beleriand, a north-western coastal area of Middle-earth.
When the Years of the Sun began, Arda experienced the so-called Second Spring, a new period of growth and vitality as a result of the light of the sun in Middle-earth. At that time, Men also awoke in the east of Middle-earth.
At the end of the First Age, Beleriand was destroyed during the War of Wrath, and much of it sank beneath Belegaer; only a few high points, such as Himring (later Tol Himling) and parts of Dorthonion (later Tol Fuin), remained above sea level as islands.
In the Second Age, Númenor was raised in the Great Sea for the Edain. This island existed through most of the Second Age, but was destroyed as a result of the pride of the Númenórean people in defying the Ban of the Valar and sailing to Aman in the west.
During the downfall of Númenor, Arda was made round. The Undying Lands (Aman and Tol Eressëa) were taken out of the world, and could only be reached by the Elves, following the straight road that was granted to them. As Aman was taken away from Arda, new lands and continents were created.
South of Middle-earth was the Dark Land, and east of it was the Land of the Sun.
Allusion to real-world geography
Tolkien stated many times that Arda was our world in a fictional time, so we can interpret the known regions of Middle-earth as Europe. Tolkien stated that The Shire was situated at roughly the same latitude as the Midlands of England, while Minas Tirith in Gondor was at the same latitude of Florence, putting Mount Doom and Mordor in the general region of Asia Minor. However, the entire continent of Middle-earth extends beyond the regions known to Gondor, far into the uncharted East and South, and Middle-earth probably encompassed all of what later became Eurasia. Information regarding both was very vague. To the south were the Hither Lands including Harad, though "Harad" means "South" and while properly used for the region immediately south of Gondor. Near Harad and Far Harad probably corresponded to North Africa and sub-Saharan Africa, respectively. There were many stretches of sun-scorched desert in Near Harad, but like Africa there was supposedly a jungle beyond it which few had ever ventured to, from which the great Oliphaunts were found. Tolkien also said that the Haradrim loosely corresponded to the Berbers (though "Haradrim" refers to a group of many races, not just the one). Men of Far Harad are also described as being black skinned. The vast eastern lands were collectively referred to as Rhûn, and like Harad they are uncharted. The Easterlings, like the Haradrim, were a diverse collection of many races, ranging from nomadic steppe horsemen akin to Turkish and Mongol tribes, to civilized and rich kingdoms stretching across the vast Eastlands. Rhûn supposedly stopped at another ocean at its eastern end. In ancient times the great chain of the Red Mountains, the eastern counterpart of the Blue Mountains of the west, ran north to south near here (all of the world was once symmetrical at the dawn of time). The Elves first awoke in the east, though their original home of Cuiviénen doubtless no longer existed by the Third Age. It is not known if the Red Mountains themselves survived into the Third Age, either intact or in some reduced form (like the Blue Mountains). The lands of the east probably collectively corresponded to the Asian continent, if Harad was taken as Africa. It is said that the Blue Wizards went far into the east of Rhûn and beyond to stir up rebellions against Sauron in the lands he held sway over, though the tales of these struggles never reached the west in detail. It is also said that the Blue Wizards somehow failed in their mission; this is interpreted several ways, either that like Saruman they set themselves up as lords of men, or that like Radagast they started to care more for the local people they were charged with protecting than Middle-earth as a whole and then refused to return. Tolkien did state that many of the "magic cults" and orders that exist in Asian countries today can trace their origins back to the Blue Wizards teaching local peoples of the east the magical arts.
There was also a separate continent south and east of Middle-earth called the Dark Land, which could correspond to Australia. Another separate continent to the east of Middle-earth was the Land of the Sun, so called because when the world was flat the Gates of the Sun were near there, and it would be scorched. After the Fall of Númenor, Arda is made round and new lands are created east of Middle-earth (and west too, now that the world is round and can be circumnavigated), out of these Sun Lands that are apparently meant to be North and South America, though none of Tolkien's mythology deals with what happens there. The Sun Lands were known to contain a north to south running mountain range which could correspond to the Rockies and Andes mountain ranges.
The original world as built by the Valar was referred to as Arda Unmarred. This world was effectively destroyed by the wars of the Valar and Melkor, and only Valinor still resembled it in part. Arda Unmarred was a symmetrical world with everlasting light.
Arda Marred was the name given to the world as it is: the world after the wars of the Valar and Melkor, and the dispersing of Melkor's Fëa and hröa throughout the world. It is this world from which are formed the Fëa and hröa of the Children of Ilúvatar, and therefore it is at times a cruel and evil world with plagues, extreme colds, heat, and other concepts which do not exist in Arda Unmarred.
Arda Marred also broke the design of Elvish immortality: in Arda Marred Elves slowly fade, until at last they are little more than wraiths. Only in Valinor was this fading delayed, which is one reason all Elves have no choice but to sail to Valinor in the end. One of the special abilities of the Rings of Power was that they could delay the effects of time, and as such were used by Elrond and Galadriel to preserve their realms. After the destruction of the One Ring in the Fourth Age, the Elvish rings' power over time was also lost, so the last of the Eldar had no choice but to leave Middle Earth and sail west for Valinor.
Arda Healed is the prophecy that Arda Marred will be restored: it will be like Arda Unmarred but better, since it will also incorporate all the good things of Arda Marred. Arda Healed will be created after the Dagor Dagorath, when Morgoth will finally be destroyed, and the world broken. Afterwards, a new world will be created and will be known as Arda Healed.
Galadriel's words while parting with Treebeard at Isengard in The Return of the King are "Not in Middle-earth, nor until the lands that lie under the wave are lifted up again. Then in the willow-meads of Tasarinan we may meet in the spring. Farewell!". This might imply that Beleriand will be recovered after the Dagor Dagorath. It is however unknown, since Arda Healed will contain all the good things of Arda Marred, whether Númenor will also be recovered or not.