A positronic brain
The positronic brain was an extremely sophisticated computation device capable of artificial sentience, created by Dr. Noonian Soong, based on an idea from author Isaac Asimov. (TNG: "Datalore")
This device consisted of an artificial neural network, designed to imitate the humanoid brain. The construction of a positronic brain was extremely complex, and Dr. Soong was the only scientist to have done so successfully – on at least seven occasions: three unspecified prototypes, B-4, Lore, Data, and Juliana Soong (although the latter three were the only truly stable units).
In early 2366, Data allowed nanites to enter his neural network in order to communicate verbally with the crew of the USS Enterprise-D. (TNG: "Evolution")
Later that year, Soong's creation, Data, attempted to emulate his father's work, and created the android Lal. However, Lal's positronic net suffered a cascade failure, and ceased functioning. (TNG: "The Offspring")
On Omicron Theta, Dr. Soong experimented with scanning the synaptic patterns of the colonists' temporal lobes and programmed them into Data's neural nets. (TNG: "Silicon Avatar")
In early 2368, Data's positronic links were severed, cutting off his brain from the rest of his body. (TNG: "The Game")
A plasma shock was enough to overload Data's neural net (TNG: "Power Play"), while a Kedion pulse would cause Data's subroutines to reset to normal if they were altered by outside sources for some reason (TNG: "Descent, Part 2").
Around 2370, Data began a discussion with Deanna Troi about the long-term effects of space travel on his positronic net, which Troi was later disinclined to continue. (ENT: "These Are the Voyages...")
Medical uses
Dr. Julian Bashir implanted Bareil Antos with a positronic matrix to keep him alive after part of his brain stopped functioning. Bareil awoke but said that the sensations he was experiencing did not seem real and seemed like a distant memory. Later, when other parts of Bareil's brain stopped functioning, Bashir chose to let Bareil die rather than replace the rest of his brain with positronic implants. (DS9: "Life Support")