While building the realm of Asgard, the gods needed help building a wall for protection. A Jötunn offered to use his horse, Savadifari, to build the walls for Asgard and in return, he would be allowed to marry Freya, the most beautiful goddess of the Aesir.
Odin and the other gods agreed, believing he couldn't do the task within the timeframe. Freya was upset because Odin didn't ask her or consider her opinion on the marriage she would have been 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 so Odin came up with a plan to get the horse distracted so it couldn't finish the job. The plan was to have Loki shape-shift into an alluring mare. The horse did get distracted by Loki and couldn't finish the job, thus Loki would eventually bear the fruit of his trickery: Odin's prized 8-legged horse, Sleipnir.
The home of the Aesir is where the great hall of Valhalla is located, the palace where the Einherjar live and train, and most 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 it 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, that finally brought peace to Asgard and the rest of the Nine Realms.
During Thor's genocidal campaign against 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 man made 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.
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 Helheim 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's rune is Gebo, which translates to "gift".