Good answers - this is just a little bit of further elaboration or restatement, since it can be confusing to a beginner.
"Compression makes loud parts softer and soft parts louder" is a bit of an oversimplification, and not entirely accurate.
As has been stated, when you compress a signal you are making some or all of the signal SOFTER. If your threshold is set high, it will only decrease the level of the very loudest parts. If the threshold is set very low, everything will come out at a lower level.
Attack time also figures into this: if the attack time is long, the faster transients will pass through unaffected and undiminished in volume. A very short attack time will capture/attenuate all or most of the transients.
Your ratio determines how much the signal is diminished once it crosses the threshold. For example, a 5:1 ratio means that if the original signal increases 5 db, after engaging the compressor that same signal will only increase 1 db. When you get up to ratios of 20:1 or more, it is considered limiting, since almost any increase in volume will produce a negligeable change after compression. Usually limiters are set with a high threshold, otherwise you would be removing all dynamics from the material.
If you look at the controls on your compressor there is usually one called "make-up gain". This is where the "soft parts louder" comes in. After using a compressor to compress (or limit) audio material, you end up with a "softer' result and less total dynamic range. The makeup gain lets you then raise the entire result - the entire track gets louder by the amount of makeup gain you apply.
BUT...since you have first applied compression, the loudest parts are now closer in dynamics to the softest parts. This not only creates available headroom to raise the entire track, but gives this result: the very loudest parts are actually raised back (with makeup gain) to THE SAME level as they were before (assuming they were at or near max volume), but since the softest parts are now closer in level to the loudest parts, they are dragged up in volume to a higher level than they were originally. This makes the average level of the track sound hotter, as well as keeping the softer parts from being overwhelmed or hidden by other tracks in the mix.
I've probably just made things more confusing. Sorry!
