The Outrealm (異界 Ikai lit. Other World in the Japanese version) is a general area in Fire Emblem Awakening that is accessible using the Outrealm Gate and by purchasing DLC.
The Outrealms are numerous areas outside of the main world of Awakening that are mostly unrelated to the main storyline and provide extra challenges for the player. In most maps, the spirit of legendary heroes and villains from previous Fire Emblem games, known as Einherjars, are encountered, requiring the player to either work with or defeat these spirits. After completing most maps in the Outrealms, the player is rewarded with an Einherjar card containing the spirit of a legendary past Fire Emblem character including Marth, Eirika, Alm, and Katarina to be recruited into their army. Other maps are used for acquiring funds, listening to special conversations at a festival, saving the future children from Grima in an alternate future, or other such missions.
Some of the realms are original locations that include a village inside of a ruin, a volcano, and even a beach. Other locations are recreations of past Fire Emblem maps like Talys from Fire Emblem: Shadow Dragon and the Blade of Light and Fire Emblem: Shadow Dragon, Gebal Castle from Path of Radiance, and the Dragon's Gate from Fire Emblem: The Blazing Blade.
In Future Past 3 and Hidden Truths 2, Grima and Lilith hint that those that come from other worlds do not retain as much power as they do in their original world.