The City of Asgard is the capital of one of the Nine Realms of the same name. It is located in one of the extremes of Asgard.
ASGARD ON EARTH
With Thor's return to Earth after the events of Ragnarok, he also recreated the capital city of Asgard on Earth itself, specifically on a piece of land outside Broxton, a small city west of Oklahoma City that he purchased for a flatbed truck of gold. Using the Odin-Force, Thor raised it up above the ground. It remained floating as Thor went to gather the Asgardians.
During the Siege of Asgard, its inhabitants battled to defend it against Norman Osborn and his forces. Many were killed in the battle, and in the end it was destroyed by the Void, who sent it crashing down to the ground. In the aftermath, an alliance between the Avengers and Asgardians was made. Tony Stark made plans to rebuild Asgard.
RECONSTRUCTION AND ABANDONMENT
In order to prepare the Asgardians for the battle against the Serpent, Odin remade the City of Asgard in the realm. When the Serpent was defeated, he banished all others from the realm, remaining there alone with his dead brother. He appointed Idunn, Gaea and Frigga, collectively the All-Mother, to lead the fallen Asgard which was on Earth. The fallen Asgard under the leadership of the All-Mother would be later rebuilt into Asgardia. By manipulating the Angels from Heven, Loki made available once more the passage from Midgard to Asgard, resulting in Odin's return from the now-decrepit City of Asgard.
After the Mjolnir of a destroyed universe crossed realities and landed in Old Asgard, the entire planetoid was abducted by the Collector in an attempt to claim the hammer. After investigating the appearance of this other Mjolnir and coming into conflict with the Collector, Thor used the hammer's teleportation power to teleport the City of Asgard away.
When the Mangog returned, he initially attacked Old Asgard. After defeating the wielder of the alternate Mjolnir, Volstagg, the creature was directed to Asgardia by Malekith the Accursed. Following the Mangog's defeat, which came not before Asgardia's destruction, the Asgardians relocated to Old Asgard and set out to recondition it so it could serve as their home once more.