A Digital Signal Processor (DSP) is generally defined as being a microprocessor specifically designed for digital signal processing, usually for use in real-time computing. In simpler terms, and with a view to its relationship with computer-based audio recording, a DSP is a piece of hardware that runs software plug-ins (audio effects and/or software instruments) within the computer-studio environment. The word 'accelerator' indicates that the DSP 'accelerates' the performance of the computer itself, which of course is already used for processing digital signals. It is perhaps something of a misnomer, as it is really providing additional but separate processing power, rather than influencing the performance of the computer's CPU in any way.
In physical terms a DSP accelerator is a silicon chip microprocessor, and is found either on an internal PCI or PCIe card which is slotted into the motherboard, or in a hardware shell that is connected to the computer via a protocol such as FireWire or USB.
Pro Tools HD3 Hardware from Digidesign
In both cases though, it is the computer which provides the 'front end', and also the integration of the DSP plug-ins within the recording software and audio signal chain.