Simply put, a signal processor is any electronic device that controls or changes the audio signal in some way in order to improve the sound.
The infinite number of different combinations of tone, loudness, rhythm, and timbre in the creation and reproduction of music often makes it necessary to add to or subtract something from an individual sound or group of sounds in order to make the whole song sound right.
Consider a vocal microphone in a live music situation, for example. The microphone picks up much more than just the voice. The other parts of the ensemble's music may also get picked up and muddy the sound — bass does this a lot. One solution is to turn down the bass tone control on that microphone channel, which will clean up the sound considerably.
Which types of processors you need depends on how much fine-tuning control you want over your music. Most mixers have some control capabilities, and that's enough for some people. But if there's a problem you just can't seem to correct with the mixer, then consider adding one of more of these processors. Outboard signal processors often feature multiple effects capabilities, so you can control more than just one aspect of your music from a single device.