Arda (Q."World"; S."Ardhon" or "Gardhon"; Ad."Abatta") was the Realm of Manwe inside of Eä. It incorporated the planet of Ambar with it's continents Middle-earth, the Dark Lands, and the New Lands, as well as the Undying Lands of Aman which were removed from Ambar towards Exambar in the Second Age.
The Spring of Arda
Eru stayed the early discord among the Ainur and ended the Great Music. Admonishing his servants, he shamed Melkor. The Black Enemy submitted to the will of the One, but a painful hatred lay rooted deep within his remorse. Eru forgave Melkor's transgression and took the Valar out of their fair home in the Timeless Halls and showed them Eä. Set amidst the Void, this World was theirs to enter, to mold in its final glory. When the Valar and Maiar left Eru's Timeless Halls and passed into Eä, the World was but a rough shape, like an unworked jewel waiting to be crafted into a finished masterwork. The arriving Ainur, seeking perfection and symmetry, set about sculpting Arda and arranging the Heavens. Melkor worked in ways contrary to the scheme of the other Valar. The Black Enemy sought a World of his own thought and he challenged his brethren. As they built, Melkor destroyed or perverted their work. War raged across the young lands. Eventually, though, the Valar united against their renegade brother. Melkor retreated, and with the coming of Tulkas — the last Vala to enter Eä — the Black Enemy fled from Arda, escaping over the Walls of the Night that bound the World. The hearty Tulkas earned Melkor's undying enmity but, for a time, Eä remained at peace.
The Two Lamps that lit the World
The World took shape during this, the Spring of Arda. Mountains and valleys emerged according to the scheme, and the land took on a placid balance. Two Great Lamps (Illuin and Ormal), erected on mountain-pillars in the North and South, gave light to Arda, and all within the circular Bounds of the World achieved the glory Eru sought. Where their glow was brightest, the Valar constructed their home. They called it the island of Almaren, which rested in a vast lake in the middle of the continent that formed the center of Arda. There the Exalted Ones relaxed in splendor, enjoying the marvels that they had created within the guidelines of Eru's plan.
The Fall of the Two Lamps
Arda's brief Spring was short-lived, however, for Melkor came out of exile. The rebellious Ainu slipped back out of the Void quietly, hoping to surprise and so vanquish the other Valar. Entering Eä in the far North of Arda, he set about building an unbreakable fortress. He delved a deep refuge called Utumno (Q. "The Valley of the Evil Hollow?") with the help of his lieutenant, Aule's chief Maia servant Sauron — who he seduced before coming to Arda. Raising a great barrier to those seeking to assail his lair, he created the Iron Mountains, a semicircle of peaks that reached across Arda's sole continent. Signs of Melkor's return began to plague the land. Healthy forests withered, and wretched quagmires appeared; foul beasts preyed upon fair fauna, and a chilling cold gripped the North. The Valar awoke from their repose and searched Arda for the Black Enemy's place of hiding. Before his discovery, however, Melkor struck a blow that ended the Spring of Arda. Leaving Utumno to his servants, he struck down the mountain-pillars that supported the lamps Illuin and Ormal, spilling their fire upon the land and ripping the continent asunder. The World fell into darkness as the seas swelled and Almaren was destroyed. Fair Arda changed, the shape of its landscape forever marred. Tulkas gave chase to the rogue Vala, but Melkor returned safely to his hold amidst the cataclysm. Reinforcing its defenses, he awaited the Valar's anticipated retribution. This revenge would not come soon, though, for the Valar turned instead to the work of restoring the land and building a new home.
Arda as our World
Tolkien stated that the history of Arda was supposed to take place in a period of the actual Old World of this planet, although in a in a 'mythical' time. He imagined the gap between the Third Age and now to be about 6000 years, our present time corresponding either to the end of the Fifth Age or actually the end of the Sixth Age or the early Seventh Age.
Many readers interpret this as the history of Middle-earth being set in a pre-historic Europe (or Eurafrasia). Another possible interpretation would be that the 'mythical' time Tolkien referred to could be seen as a possible alternate past, so the history of Middle-earth would be set in the actual Old World of this planet, but not in our real history but in a different (or alternate) reality.
It was unknown if there were (or are) other Worlds aside from Ambar or Aman. Tolkien stated that "… We cannot say that there ‘must’ be elsewhere in Eä other solar systems ‘like’ Arda, still less that, if there are, they or any one of them must contain a parallel to Imbar.…" but "… The demonstration that there existed elsewhere Incarnates, parallel to the Children of Eru, would of course modify the picture, though not wholly invalidate it. …". Tolkien did not explicitly deny the possible existence of other Worlds, but he also didn't include any other Worlds into his cosmology, with the possible exception of Ellor, Emberû, and Tekel-Mirim, worlds, probably different planets, that were explored in lucid dreams by members of the Notion Club. Tekel-Mirim was a crystal-world removed far from Earth in space and time. One possibly alien race were the elf-like but non-incarnate En-Keladim.
The Silmarillion stated that Tulkas came to Arda from other regions of Eä, and that Melkor withdrew (from Arda) and departed to "other regions" and did there what he would. This could be interpreted as possible hints to other worlds within Eä, where incarnate Ainur existed, or other Worlds which were visited by Melkor during his exile. However, even if there had been, or still are, other Worlds beside Arda or Ambar, the Eruhini of Arda knew nothing of these or had little interest in them.
Outside of the Canon, there are allusions to other Worlds. The planet Kethira (the setting of Hârnworld) is implied to be one of seven linked parallel worlds collectively known as Keléstia, the other six worlds (or Planes) are said to be Terra (or Earth), Yàsháin, Midgaad, and the Blessed Realm. The two latter of which are implied to correspond with Middle-earth (or Ambar) and Aman. Similarly Shadow World or Kulthea (Setting of Loremaster and Rolemaster), could be in many ways seen as a world parallel to Arda or Ambar.
It is also mentioned that the star Carnil (Gormok) is actually Mars. Other planets within Eä mentioned are Jupiter (Alcarinquë or Silindo), Luinil (guessed to correspond with Neptune, Uranus, blue Rigel, Spica, or Regulus), Mercury (Elemmíre), Neptune (Nénar), Venus (Earendil’s star, Rothinzil, Gil-Estel, Zingil), Saturn (evtl. Lumbar, Eneköl, Gyürüchill, Shomorú ?), and eventually Tengyl or Tannacolli and Til.