Compression is something I am trying to learn more about, but admittedly it is still somewhat mysterious to me. Is compression when playing live a common thing? Would you apply it in your signal chain (e.g. bass to compression pedal to amp / FOH) or would it be applied by the engineer for FOH? Or both?
I think the issue is that there are many, many types and uses of compression, and they get conflated. There's the old definition that goes something like "loud stuff gets quieter and quieter stuff gets louder", but since the uses are so varied, this definition causes a lot of confusion, and a lot of flat signals robbed of all dynamics by unfamiliar users.
Think about a tube amp set to a particular level. With your instrument volume rolled almost all the way down, you'll hear a quiet, clean signal. Roll the volume up to halfway and you'll hear the same clean signal, only louder. Roll it up 3/4 of the way and it might be louder, but slightly distorted. Roll it up all the way and it might be only slightly louder than 3/4, but even more distorted, so you
perceive it as much louder. This is compression happening at the preamp stage of your amp, and it is a form of
tone shaping.
Imagine you're in a studio. You just recorded your clean bassline to a console, and everyone in your band finished their parts up as well. Picture the entire finished song as a waveform: louder parts have larger spikes, while quieter parts have smaller spikes. Your engineer prints this waveform to tape, and every time those larger spikes hit the edges of the tape, they're cut off dramatically. Those instances where the waveforms clip will express themselves as a kind of distortion. That's also compression.
Most players use compression pedals on their boards to shape the transient of their playing. When your fingers or your pick hit the strings, they immediately make a loud "click" sound, known as the
transient, followed by a quieter
sustain, or ringing out of the string. The compressor allows you to adjust the volume at which the compressor kicks in (threshold), the amount that it lowers the transient (ratio), and if you have an attack knob, the speed at which it clamps down on the transient. This gives you a lot of flexibility to shape your tone.
Additionally, some players use a compressor as a
limiter. This is more a means of control, so that if your signal gets too loud, the limiter will clamp down hard, at a very high ratio, so that your peaks don't blow out the speakers or the audience's ears. This is also how most FOH engineers will use compression, since they're less interested in shaping your tone and more interested in protecting the gear/maintaining a reasonable volume.
The best thing to do is get a cheap compressor pedal or free VST and push it to the extremes: low threshold, high ratio, fast attack. As you pick your bass, gradually increase the attack and listen for that click - you'll hear it change dramatically as the attack time slows down. That's compression happening in real time.