Perhaps the most benign side effect is that of added 'warmth', but go too far and you can introduce audible pumping, which doesn't usually sound that great unless you need it as an effect. It's when you're processing vocals that the subtle differences between compressors come to light.
Some units attempt to do both, but few are entirely successful.
As a rule, if you want to make a transparent compressor produce more of an obvious effect, you can increase the ratio and at the same time shorten the release time. The shorter the release, the more obvious the gain pumping effect of the compressor. As you make these adjustments, watch the gain reduction meter and adjust the threshold control to get the amount of gain reduction you need. If you don't have one, set the attack to its fastest, and the release time to between 300ms and half a second. Before moving onto other areas of application for compressors, I need to mention noise.