Digital signal processing (DSP) is concerned with the processing of digital signals or analog signals after converting from analog to digital format.
DSP includes subfields like: communication signals processing, radar signal processing, sensor array processing, digital image processing, etc.

DSP is usually used with real-world analog signals found in our lives, the first step is usually to convert the signal from an analog to a digital form, by using an analog-to-digital converter. Often, the required output signal is another real-world analog signal, which requires a digital-to-analog converter.

Digital signal processing algorithms can run on:
 General purpose microprocessors and standard computers.
 Specialized processors called digital signal processors (DSPs).
 Purpose-built hardware such as application-specific integrated circuit (ASICs) and field-programmable gate arrays (FPGAs).
 Digital Signal Controllers (mostly for industrial applications such as motor control).
 Stream processing that works very well for traditional DSP or graphics processing applications (such as image, video).

Related fields 
 Automatic control
 Computer Science
 Data compression
 Electrical engineering
 Information theory
 Telecommunication

References

Other websites
Free Video Tutorials on Digital Signal Processing by Prof. Dutta Roy

Further reading 
Alan V. Oppenheim, Ronald W. Schafer, John R. Buck : Discrete-Time Signal Processing, Prentice Hall, 
Boaz Porat: A Course in Digital Signal Processing, Wiley, 
Richard G. Lyons: Understanding Digital Signal Processing, Prentice Hall, 
Jonathan Yaakov Stein, Digital Signal Processing, a Computer Science Perspective, Wiley, 
Sen M. Kuo, Woon-Seng Gan: Digital Signal Processors: Architectures, Implementations, and Applications, Prentice Hall, 
Bernard Mulgrew, Peter Grant, John Thompson: Digital Signal Processing - Concepts and Applications, Palgrave Macmillan, 
Steven W. Smith: Digital Signal Processing - A Practical Guide for Engineers and Scientists, Newnes, 
Paul A. Lynn, Wolfgang Fuerst: Introductory Digital Signal Processing with Computer Applications, John Wiley & Sons, 
James D. Broesch: Digital Signal Processing Demystified, Newnes, 
John G. Proakis, Dimitris Manolakis: Digital Signal Processing - Principles, Algorithms and Applications, Pearson, 
Hari Krishna Garg: Digital Signal Processing Algorithms, CRC Press, 
P. Gaydecki: Foundations Of Digital Signal Processing: Theory, Algorithms And Hardware Design, Institution of Electrical Engineers, 
Paul M. Embree, Damon Danieli: C++ Algorithms for Digital Signal Processing, Prentice Hall, 
Anthony Zaknich: Neural Networks for Intelligent Signal Processing, World Scientific Pub Co Inc, 
Vijay Madisetti, Douglas B. Williams: The Digital Signal Processing Handbook, CRC Press, 
Stergios Stergiopoulos: Advanced Signal Processing Handbook: Theory and Implementation for Radar, Sonar, and Medical Imaging Real-Time Systems, CRC Press, 
Joyce Van De Vegte: Fundamentals of Digital Signal Processing, Prentice Hall, 
Ashfaq Khan: Digital Signal Processing Fundamentals, Charles River Media, 
Jonathan M. Blackledge, Martin Turner: Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications, Horwood Publishing, 
Bimal Krishna, K. Y. Lin, Hari C. Krishna: Computational Number Theory & Digital Signal Processing, CRC Press, 
Doug Smith: Digital Signal Processing Technology: Essentials of the Communications Revolution, American Radio Relay League, 
Henrique S. Malvar: Signal Processing with Lapped Transforms, Artech House Publishers, 
Charles A. Schuler: Digital Signal Processing: A Hands-On Approach, McGraw-Hill, 
James H. McClellan, Ronald W. Schafer, Mark A. Yoder: Signal Processing First, Prentice Hall, 
Artur Krukowski, Izzet Kale: DSP System Design: Complexity Reduced Iir Filter Implementation for Practical Applications, Kluwer Academic Publishers, 
John G. Proakis: A Self-Study Guide for Digital Signal Processing, Prentice Hall,