(covers information from several alternate timelines)
An engram or memory engram was an impression made physically in neural tissue of the brain by any mental stimulus, thereby explaining the persistence of memory. The act of deliberately storing a memory was known as memorization.
Shortly before 2268, Dr. Richard Daystrom developed multitronics – a method of constructing a computer that could think, and of impressing memory engrams upon its circuits. The result was the M-5 multitronic unit, developed with Daystrom's own engrams. As Daystrom was near a nervous breakdown at the time, the M-5 was (like its predecessors M-1 through M-4) "not entirely successful". While multitronics showed some promise, it was eventually superseded by isolinear circuitry. (TOS: "The Ultimate Computer")
The Banea punished murderers by forcing them to relive their victim's last moments every fourteen hours; this was done by inserting the memory engrams of the victim into their brain. Tom Paris' reaction to this method showed that it harmed the human brain. (VOY: "Ex Post Facto")
The Mari sought out memory engrams that contained the memories of violent thoughts. The Mari then removed those engrams, because having violent thoughts was a crime on their world. This procedure was known as an engramatic purge. (VOY: "Random Thoughts")
In 2375, several of Data's memory engrams were damaged after he was shot by a Son'a weapon on the Ba'ku planet. This caused the activation of his memory loss fail-safe system. Later, on board the USS Enterprise-E, Geordi La Forge replaced them. When Data was reactivated, his first inquiry was pertaining to the missing engrams. (Star Trek: Insurrection)
In an alternate 2379, Kes had lost over ninety-nine percent of her memory engrams due to the effect of the morilogium. After being sent back in time a few weeks due to a side-effect of The Doctor's bio-temporal chamber, she had lost over ninety-eight percent of the engrams. After yet another such time-shift, to her ninth birthday, the amount was reported as over ninety-five percent. (VOY: "Before and After")
In the 29th century, Lieutenant Ducane explained that because Kathryn Janeway and Seven of Nine were never transported into their relative futures, their memory engrams would not require re-sequencing. (VOY: "Relativity")