Part of the "Megastructures: The Visual Encyclopedia" book project, on sale now at http://www.neilblevins.com/megastructures/megastructures.htm
Updated from an older piece of artwork.
A Matrioshka Brain would be a type of Dyson Sphere whose purpose was to power an enormous supercomputer. The concept was invented by Robert Bradbury in the anthology Year Million: Science at the Far Edge of Knowledge. The name comes from the Matryoshka doll, a type of wooden nesting doll made in Russia.
A Matrioshka Brain would involve nesting several Dyson Spheres inside one another. The innermost Dyson Sphere would collect power directly from the sun, then use that power to run the computer, and in the process give off waste heat. That heat would be absorbed by the second Dyson Sphere, which would use it to power its own supercomputer. The waste heat from the second sphere would be captured by the third, etc. Each sphere would receive less and less power as the waste heat decreased as it worked its way out from the star. In an ideal situation, the final outermost Dyson Sphere would be close to the temperature of interstellar space, which would mean you had used all of the sun’s energy without any waste.
The computer could be used for any number of purposes, such as simulating universes or storing trillions of emulated human brains. The computer could be a single giant computer, in which case you’d have to worry about lag transmitting data between the layers, or you could use a series of separate computers, one per shell.