Asgard is one of the Nine Realms of the World Tree and the home of the Aesir Gods.
Norse Mythology
While building the realm of Asgard the gods need some help building wall for protection. A Jötunn offered to use his horse Savadifari to build the walls for Asgard and in return he would marry Freya the most beautiful goddess.
Odin and the others agreed believing he couldn't do. Freya was upset because Odin didn't ask her how she felt about the marriage that she would be forced into. Freya also didn't like that Odin and the other gods tricked and lied to the giant.
They became worried when it looked like he might complete the job Odin came up with a plan to get the horse distracted so it couldn't finish the job. The plan worked Odin had stopped the giant from fishing the task with the help of Loki.
Description
The home of the Aesir is where the great hall of Valhalla is located, the palace where the Einherjar live and train, and must likely the place where Odin's court gathers as well. Asgard was also one of the battlefields in the Aesir's raging war against the Vanir, thus was ravaged after the centuries-long war.
It was the marriage of Odin, the King of the Aesir, with Freya, one of the leaders of the Vanir, what finally brought peace to Asgard and the rest of the realms.
During Thor's genocidal campaign of the Jötnar, Asgard's walls were in need of fortification, but none of the Aesir wanted to take care of such a menial task, until one day, a strange manmade a bet with Odin. In exchange for reinforcing Asgard's walls in a near impossible timespan, the stranger, actually the Jötunn Hrimthur in disguise, would be allowed to talk to Freya. According to Mimir, Hrimthur hid a weak spot in the walls' defenses and entrusted the secret only to Freya, before being discovered and killed by Thor.
After the marriage's fallout, Odin locked the path to Asgard through Týr's Temple, allowing nobody except for those who Odin allows to visit the realm.
The God Realm was the place Baldur attempted to take Atreus and Kratos to when he kidnapped the former, but Kratos interrupted the transportation sequence of Týr's Temple, thus landing in another realm instead of Asgard.
Asgard itself hasn't been shown so far, but pieces of the realm's materials known as "Asgardian Steel" can be collected after defeating the corrupted Valkyries hidden through the Nine Realms.
Asgard is destined to burn at the hands of the Fire Giant Surtr when Ragnarök comes.