Great Black Eye

The Great Black Eye, (PROSE: The Adventuress of Henrietta Street, The Domino Effect, The Gallifrey Chronicles) also known as the sun-god of the Babewyn and called the Opposition by Sabbath Dei, (PROSE: The Adventuress of Henrietta Street) was a powerful and fearful force which watched the post-War universe from somewhere outside of time. (PROSE: The Adventuress of Henrietta Street, The Gallifrey Chronicles)

Sightings of the Eye
At the end of the conflict between tigers and humans on Hitchemus, a hole briefly appeared in the clouds which "looked like an eye in reverse, blue pupil and black iris," staring down at the climactic events. (PROSE: The Year of Intelligent Tigers) In a mirroring of Pablo Picasso's Guernica, a blazing eye watched over the destruction of the System, illuminating the chaos as the System blackened and fell to greyness. (PROSE: History 101)

The Oracle witnessed the Great Black Eye as one of the terrifying creatures outside of reality, and believed this Eye was what was forcing many powerful forces, such as the Oracle, (PROSE: The Domino Effect) Time Wraiths, (PROSE: The Slow Empire) and clock people, (PROSE: Anachrophobia) to flee the Time Vortex. It believed that the collapse of the multiverse and the vortex would fracture the barriers which normally kept the Black Eye from interacting with normal reality. When the Oracle met the Eighth Doctor, it knew that he had seen the Eye before and tried to make him remember the horrific nature of the Eye so that he would support its plan. (PROSE: The Domino Effect)

Nature
By several accounts, the Black Eye was speculated to be a corrupted version of the Eye of Harmony, a power source of the lost planet Gallifrey. (PROSE: The Adventuress of Henrietta Street, The Gallifrey Chronicles) The Eighth Doctor's second heart was "the organ through which he was watched by his home's Great Eye, by which the powers of the elemental world were passed to him," but with Gallifrey gone the Doctor's second heart nearly killed him from pumping his body full of the disease and darkness of the Kingdom of Beasts. (PROSE: The Adventuress of Henrietta Street)

However, the Doctor later believed that his illness had been caused simply because his heart hadn't been linked to anything, and that the only Eye of Harmony still in existence was the independent power source of his own TARDIS. When the casing of the TARDIS' Eye was cracked open by an atomic explosion, the Doctor found himself being watched by the great Black Eye of the Master, who watched over the universe with god-like power from within the heart of the TARDIS. (PROSE: The Gallifrey Chronicles)

Per another account, the black eye belonged to a Dalek which watched, waited, and planned somewhere beyond the Time Vortex and the multiverse, monitoring reality for what seemed like forever. (PROSE: Sometime Never...) In the post-War universe, the Eighth Doctor had several glimpses of Daleks at the edges of his adventures. (PROSE: Father Time, The Last Resort) After the destruction of the Council of Eight, the Dalek began tracking Soul and Zezanne as they fled with the star-killer and became stranded in London 1963. (PROSE: Sometime Never...) The equivalents of Soul and Zezanne in the prior main timeline notably attracted Daleks to London 1963 when they brought the a star-killing weapon there. (TV: Remembrance of the Daleks)

Behind the scenes

 * As clarified in an interview with Justin Richards, the original intention of the recurring Black Eye was that it would be the eyestalk of a Dalek. While the Dalek storyline and reveal did not come to fruition the brief cameo by a Dalek at the end of Sometime Never... confirms this connection in-universe.