Compression and Dynamics Processing are essentially the same thing, so if you see either term used in an article, you'll know they are the same and not some devious way to confuse you.
The fundamentals of compression (this is what I'll be calling it for the purposes of this post), are a bit difficult to grasp when starting out.
Every single recording you hear uses compression in some way or other, in fact radio stations use it so much that people have subconsciously come to expect it, even if they don't know what it is.
The way that I remember how compression works, is by thinking of listening to a recording of a meeting or lecture of some sort, led by a speaker or lecturer for half an hour or more.
Imagine that the speaker keeps speaking very loudly and then, without warning, very quietly; it does happen. It would be nice if he could somehow speak at a more even volume throughout.
The loud bits are known as peaks and the quiet bits are known as troughs, think of a line graph, if you will. These variations in volume are known as dynamics, hence the dynamics processing term when you use equipment to, well, process them.
The object is to even things out so that the loud bits are a bit quieter and the quieter bits are a bit louder. If you were the person recording the lecture, you would probably use a mixer of some sort with faders. Imagine using the fader used to control the microphone level as the you record the lecturer, you would have to push the fader up a bit every time he/she spoke quietly and pull it straight back down again when they speak more loudly, to get a more even recording to tape or disc.
A compressor or dynamics processor effectively does this for you and because it's a machine it responds to the peaks and troughs more quickly than your human ear and reactions can.
A typical compressor unit has, at its most basic level, a threshold control, ratio control, release control and output level control.
The threshold control tells the compressor at what level
to start compressing or, if you like, move the fader down. It won't do anything until that volume reached.
The ratio control tells the compressor how much
to flatten the peak by when it occurs, a higher number means more flattening. This is how much you would bring the fader down from its current position.
The release control tells the compressor when to let go
and stop flattening the peak, or push the fader back up when the volume is at a certain level.
And the output control is the overall level that the compressor sends the sound back out at.
If you apply compression to a whole recording, like the lecture, you will find that it is easier to listen to as the ears get more tired when trying to listen to something with many fluctuations in volume, especially over a long period of time. This is one reason for radio stations using compression, as many people listen for long stretches; at work for instance.
As a producer, you can use compression on anything that needs it within the mix, such as individual instruments, and you can use it on the whole mix.
There's a whole lot more you could say about compression, but just experiment with compression principles in mind and you shouldn't go far wrong.