Of course, EQ-ing takes up majority of getting a clean mix, but it seems like you're looking for the extra 10% to clean your track up.
Compressors can reduce the dynamics of an instrument and can get one to sound smoother, punchier, or even just plain louder. There is usually a pre-EQ, then a compressor, and a slight post-EQ. As in terms of cleaning up a mix, they can really do that by stabilizing crazy dynamics, such as the attack of a clean guitar.
As far as the parameters go and how to apply them on specific instruments, that's something we need tangible examples for. There are always general guidelines for specific instruments, but in this day and age, there is so much more room to do things in different ways.
The threshold is the level of where the compressor will start to work--it anything above that threshold line will be compressed. The Ratio will decide how much everything above the threshold will be compressed. For example, a 1:1 ratio will not compress anything.
The attack is how much time the compressor will delay its activation. You can use this to shape transients (this is considered a bit old-school now that there are transient shapers out there, but still, of course, extreme viable). The release is how fast the compressor level returns back to 0db (this is a bit harder to explain without visual cues), but a faster release time will make it louder and sharper, and a longer release time will make it sound smoother and have a lot more volume ducking (provided dynamics are insane).

This is a simple formula of how it all works. Note that the output is not affected by a post-gain.