Traditionally, analog signal processing is done using analog components (e.g. resistors, capacitors, inductors, etc) which transform analog quantities (represented by voltage or current) according to the physical laws underlying the given elements. The behaviour of a circuit constructed from these elements can be analyzed, sometimes analytically but more often only numerically, especially if nonlinear elements are involved. The behaviour can be represented as a system of algebro-differential equations. This is routinely done in circuit-simulating programs (e.g. Spice). The circuit "solves" this system of equations, and presents the "solution" as the voltages at the nodes of the circuit (and the currents in its branches).
Digital Signal Processing, in contrast, represents the analog quantities as numbers (after having them digitized by an Analog-Digital-Converter). These numbers are transformed according to algorithms, producing other numbers. The stream of numbers generated this way can again be transformed back into the analog world (by using a Digital-Analog-Converter). Let me give an example: two equally valued resistors in series connected to ground, with the output signal taken from the connection in the middle. According to Ohm's law the output voltage is 0.5 times the input voltage. So if the input voltage is represented as a stream of numbers, the output voltage (again as a stream of numbers) can be computed by multiplying the elements of the input stream each with this factor. More generally, when elements are involved which store energy, i.e. capacitors or inductors, the resulting equation will be a difference equation, using the input values as well as the output values at this point in time as well as from past samples.
So all in all, Digital Signal Processing replaces analog components by algorithms, which in turn can be implemented either with specialized computing elements ("Signal Processors") or, if processing time allows, with programs running on general-purpose microprocessors.