"Positronic brain" was a term coined by Isaac Asimov. (For more information on it, just click the ol' link.) While Asimov made many useful, interesting, and important observations involving potential future robots in his various stories about them, this is probably his least important contribution in and of itself.
Because the idea of a "positronic brain" is pretty much completely insane.
"Positronic" refers to positrons. That is to say, antielectrons. That is to say, antimatter.
So a positronic brain would be, even if it was carefully shielded, full of explosive material. Not a good idea.
A further issue is the fact that, since they're antimatter, producing the number of positrons that would be involved in positronic brains would probably be hideously expensive. (If you read the positron article, it mentions that somehow, somebody's been making billions of them; the necessary number for a positronic brain, especially as Asimov defined it, would probably range towards the number of the Avagadro constant or so. That is, billions and billions of billions.)
So, yeah, Asimov was a good author, but some of his technobabble is dated. Not surprising. (He coined the term around 1940. Holy cheese.)