A digital signal processor (DSP) is a specialized microprocessor designed for high-speed mathematical manipulation of digital data. Unlike most microprocessors, DSPs are heavily optimized to perform addition and multiplication -- usually by the use of Very Long Instruction Word (VLIW) technology to allow multiple operations to be compressed into a single atomic instruction. Because of this specialization, DSPs are exceptionally good at vector and matrix manipulation.
The leader in DSP technology is Texas Instruments (TI), who has been releasing DSP chips for over a decade. Their chips allow programming in assembler or C, with only basic library support. There are other players entering the game, however, such as Philips Semiconductor with their Trimedia chip, which is optimized for consumer video applications and includes libraries with support for MPEG-2, motion JPEG, and video manipulation, supporting data on a frame-by-frame basis as well as a streaming basis.
Some consumer where DSPs may be found are cell phones, DVD players, and sound cards, though they also have large use in complex control systems due to their matrix manipulation abilities.