According to Asimov:
"The miles of relays and photocells had given way to the spongy globe of platinum iridium about the size of the human brain."
Little more is said about the operation of the robotic brain other than the implication that it somehow relies on pairs of positrons and electrons being created and annihilating each other. We are however told that the production of a single positronic brain requires more than 75,000 operations.
Asimov was far more interested in the "firmware" of the positronic brain - the famous three laws of robotics which were built into every robot at a fundamental level.
The concept of a positronic brain, although undefined and meaningless, has become so well accepted that it has appeared in the work of other writers. Perhaps the most famous modern positronic robot is Data from Star Trek TNG.