Signal processing devices are units used in a live setting to manage audio routing and processing. These units take care of the functions not covered by mixers or amplifiers.
Some of these units (such as DI boxes) condition your inputs, preparing them for a mixer. Others prepare the signal for amplification (such as Compressors, Limiters, Equalizers and Exciters). Others still will direct audio, splitting and directing it to the correct destination (crossovers and loudspeaker management systems).
Also in this category are utility devices, great for solving some of the problems associated with live sound such as groundloop isolators and feedback destroyers.