The Abyss is the deepest level of the Astral Plane, called "Hell" or the "Void" by some. This realm is the polar opposite of the World of Ideas, where souls that have accepted death without any lingering regrets or attachments eventually pass on to.
Not much is known of the Abyss save that it's located underneath the realm of the God Hand and is the home of the Idea of Evil. Because of the nature of this layer, any form of individuality is purged, as the spirits of the dead all form an interconnected chain. The chain usually appears whenever an Apostle dies, emerging from the death wound to cover the body before retracting with the Apostle's soul, while the body is reverted to its original human state. While Griffith first encountered a fraction of them while incarnated in the Tower of Rebirth, Guts first sees the chains after Wyald's death. Later on, he would become more familiar with the Abyss during his battle with the Count as the chains attempt to take him twice due to his Brand of Sacrifice.
It does not appear to be Hell in the traditional sense, as in, a place of punishment or torment, but rather a place where souls are lost in the amalgam of space and matter within it. Those that make sacrifices to the God Hand, and others who deal with demons (which usually includes the victims of those same Apostles) are taken here when they die.