disclaimer: I'm not good at music, take with a grain of salt and prefer other info
In general, I use compression for two reasons:
flattening the dynamic range of a track/group of tracks/vst/vocal/recording
mastering
Since I guess you are interested in mixing, I'll just explain 1.
There are two kind of dynamics you might want to tame with a compressor (other than drums, and that is still kind of black magic to me as to how it works exactly, I tend to try to start with the right samples).
The first type is the kind of dynamics created by a person singing or a bassist playing. Some words might be louder, some softer, some away from the microphone. If you make the level of the recording so that you can hear the loudest parts, the quietest parts will not be heard. If you bring the level up so that the quietest parts are heard, the loudest parts will burst your eardrum :P
The second type may come from a synth. You may have the decay of the synth set to very fast, and it overall sounds good. However, in the first milliseconds of the notes, you will have spikes in volume much bigger than the rest of the sound. You might want to suppress just those peaks which might not even be heard, both to create "headroom" (less peak level in the whole track), or because you might want to have a softer sound.
Note that the examples I gave for the two types (more "long term" dynamic differences and very "short term" dynamic peaks) are just examples, you might get the first type from a synth that you're playing with a keyboard and is velocity sensitive... they're just examples to illustrate the two use cases.
For the first type, you would use relatively long attack and release. You usually don't want to hear anything starting to distort. Try to put a compressor with 0.1ms attack and 0.1ms release on a vocal, bring down threshold and up ratio and listen to how horrible it sounds. I usually bounce/freeze/render the track I'm compressing to check that the attack and release are short enough, that treshould is down and ratio up enough that the track is getting compressed. Use your ears and doublecheck with your eyes :D
For the second type, you would use very short attack and release. I use many times a soft clipper instead of a compressor. Be careful about the treshold, it should only catch the peaks, and the ratio should be high.
These problems can not only present themselves just after the recording or synth, but if you do processing (eq, distorsion) you also may need compression to compensate for problems these might cause.