The Far Realm is a realm outside the Astral Sea and the Domains. At one point in time, eons ago, the Living Gate stood as a doorway between the two until it was shattered, an event which created the Shardminds. Only the power of Ioun, god of knowledge, prevented the Far Realm from completely passing into the Astral Sea.
The most coherent descriptions of the Far Realm say it is a composed of thin layers ranging in size, living, breathing, spawning and dying. Few who dare venture into the Far Realm come back alive, even less comeback with their sanity intact. Shardminds work to rebuild the Living Gate to once more cut off the Far Realm. Psionics are believed to have developed as a way to battle creatures from the Far Realm the same way a body fights infections.
Parts of the Far Realm have managed to seep into the Underdark. The cruel, ruthlessly intelligent and evil Mindflayers originated in the Far Realm. It is a place of nightmares and horrors few could witness outside of the Abyss. Many refer to the Far Realm as "the Outside" because it is literally outside the reality as most understand it.
Source: http://michaelkscott.com/dnd/