A positronic brain was a form of artificial brain.
Daleks
In 1940s, the Daleks created an android scientist Bracewell, that was implanted into the British scientific community to develop technology for the war effort. The creation was said to be controlled by a positronic brain. (TV: Victory of the Daleks)
On Vulcan, the human space colonist Lesterson examined "dead" Daleks and, thinking of them as robots, conjectured they possessed a sort of positronic brain. (TV: The Power of the Daleks)
When the Second Doctor was forced to work for the Daleks in extracting the "human factor" from Jamie's emotions, he imprinted it into a positronic brain. This was to be implanted in the Daleks, which the Daleks hoped would have made them invincible. (TV: The Evil of the Daleks)
Other references
The Dronebots used as maintenance robots on the Figaro Xll colonial supply base were provided with simple positronic brains. (COMIC: The Frankenstein Particle)
The Fourth Doctor recognized the labyrinth-like building complex that served as the lair of the Nimons as resembling both physically and functionally a "giant positronic circuit". (TV: The Horns of Nimon)
The androids of EarthWorld had positronic brains. (PROSE: EarthWorld)
Behind the scenes
A positronic brain is a fictional technological device, originally conceived by science fiction writer Isaac Asimov (1920-1992). Its role was to serve as a central computer for a robot, and, in some unspecified way, to provide it with a form of consciousness recognizable to humans. When Asimov wrote his first robot stories in 1939 and '40, the positron was a newly discovered particle and so the buzz word positronic — coined by analogy with electronic — added a contemporary gloss of popular science to the concept.