Asgard is a defensive technique that can only be used by a descendant of Finé, and only if Finé's soul has awakened in them.
It consists of a lattice of energy being created in the path of an incoming attack. However, the thing that makes it extremely tough is not how powerful a single layer is, but the fact that multiple layers can be created if necessary.
In the manga, Finé is shown using Asgard's energy to power up her moves in hand-to-hand combat when she isn't utilizing a relic.
In Episode 12, Finé would use it twice, once as a defense against a Berserk Hibiki Tachibana. However, the one layer she created was quickly destroyed. Later, when fighting against Tsubasa Kazanari, she weaved multiple layers of Asgard to prevent herself from being defeated by Tsubasa's Heaven's Wrath, unwittingly making it possible for Tsubasa to destroy her true target, the Kadingir.
In Episode 13, she would use it one last time to protect herself from an attack by Tsubasa, but at the cost of being able to keep the Durandal, and with it, her victory.
It most likely will never appear again, as Finé chose to sacrifice herself so Shirabe could survive the otherwise fatal injuries she received preventing Kirika from committing suicide upon learning that she hadn't been Finé's reincarnation at all.