Anti-time

Anti-Time was a substance theorized by the Time Lords that would break down the Web of Time. The Antiverse was comprised of it.

Description
Just as matter is opposed by antimatter, and every action has its opposite reaction, the Web of Time -- positive time -- must have its negative counterpart.

The Eighth Doctor described it thus:
 * "Anti-Time: as intractable and destructive a force to causality as antimatter is to space. Something with no past, no present, no future. A perpetuity of meaningless chaos. A Now, with no beginning or end. Elegant, brilliant, thoroughly logical... and utter gibberish." (BFA: Neverland)

Time Lord theories suggested that Anti-Time was created when Rassilon created the Eye of Harmony, binding the universe to a single continuity.

History
The theory of Anti-Time's existence was discredited until a series of anachronisms started wreaking havoc on the Web of Time. The Time Lords traced the source to the Eighth Doctor's companion Charley Pollard. Using her as a conduit, Madam President Romana and CIA Coordinator Vansell piloted a Time Station through her to the Antiverse -- a universe made of Anti-Time inhabited by Neverpeople.

The Neverpeople attempted to destroy Gallifrey using a bomb filled with Anti-Time. However, the Eighth Doctor saved Gallifrey by materialising his TARDIS around the Time Station, thus containing the Anti-Time explosion. The Doctor and his TARDIS absorbed the Anti-Time energy released. Both were "infected" by it, and took on the persona of Zagreus. (BFA: Neverland)

Zagreus was courted by Rassilon to become his assassin. With the help of Charley and the TARDIS' "good side", the Doctor managed to reassert his personality long enough to refuse the offer and cleanse the infection from the TARDIS. The Doctor, however, was forced to travel into a Divergent Universe to keep the Anti-Time and the Zagreus personality under control. (BFA: Zagreus)

The Anti-Time was eventually purged from the Doctor's system by Rassilon. (BFA: The Next Life)