Euphemism for the acrid Silicon flavored smoke that is emitted when an IC gets fried. It actually ranges in color from light blue to dark gray and is sometimes preceded by an audible pop and/or fizzle.

The theory goes that computers run on blue smoke. And when you let the blue smoke out of the chip, it will no longer work.

I had first seen this phrase used in a cartoon within a Commodore enthusiast magazine in the mid to late 1980s. The computer technician had pointed this out to the customer as to the reason why his Amiga wasn't working anymore.