The Dominus Institute was an institute set up by the Decayed Master after his mind was swapped with a future incarnation of his.
While he remembered being in great pain, the process left gaps in his memory. He proceeded to return to a site of former imprisonment to construct a plan to regain his memories. He returned to a prison in the south of England, where his former self had lured a Carmentine Mind Leach. He set up the Dominus Institute in order to lure the Doctor to him, planning to absorb his intellect. He did this by inviting various geniuses from across Earth, hoping the Doctor would notice.
The Sixth Doctor made a deal with the mind-leech, one which had the leech only take the Doctor's short-term memory. This was enough to sustain the Master but ruined his overall plans. The Master fled from the prison in his TARDIS. (AUDIO: Vampire of the Mind)