Memories were the mental record of a lifeform's past experiences.
A man is the sum of his memories, you know. A Time Lord even more so.
The Fifth and Sixth Doctors both noted that memories from before their second regeneration were vague. (PROSE: Cold Fusion, COMIC: The World Shapers) According to Madame Xing, by the time the Eighth Doctor visited her, his memory had been interfered with on thirty-seven separate occasions, with eight of them still uncorrected. (PROSE: Halflife)
Clara Oswald read the Doctor's name in The History of the Time War. When that timeline was later averted, however, she had no memory of it. (TV: Journey to the Centre of the TARDIS) The telepathic circuits of a dead TARDIS retrieved some of those lost memories. (TV: The Name of the Doctor)
Retcon was a drug used by Torchwood to remove memories of the organisation and/or alien encounters from witnesses. (TV: Everything Changes, Adam, They Keep Killing Suzie, Meat, Something Borrowed, COMIC: Shrouded) The drug was also used by other individuals and organisations, notably the mayor of the trap street, Ashildr. (TV: Face the Raven)
Because their time streams were out of sync, the War Doctor and the Tenth Doctor couldn't retain memories of saving Gallifrey along with the Eleventh Doctor. (TV: The Day of the Doctor) Because he could not remember the true outcome of the war, the Doctor suppressed their memories of their war incarnation, not acknowledging him as an incarnation.
In order to prevent Clara Oswald from dying, the Twelfth Doctor tried to use a neural block to erase himself from her memories. Upon learning of his plan, Clara reversed the polarity of the device and the Doctor lost his memories of her instead. (TV: Hell Bent) The Doctor's memories were restored near the end of his life when a glass avatar of Bill Potts allowed him to see her once more. (TV: Twice Upon a Time)
After Bill Potts's encounter with a puddle of sentient oil in the form of her friend Heather, the Twelfth Doctor intended to wipe her memories of the event. Bill realised his intentions and pleaded him not to. Eventually she gave in but before he could go through with it, she told him to imagine how it would feel if someone did it to him. The Doctor, perhaps remembering how Clara Oswald distorted his own memories, reconsidered his actions and allowed Bill to remember. (TV: The Pilot)