There are two parts to this: The spring and the spring damper (we call the dampers "shocks" in the US).
The spring is a simple mechanical device, in that by its design it provides so many ft lbs of resistance. Most springs are progressive, i.e. you can usually see two or more "zones" in the spring that offer various resistance. Usually a small portion of the spring is soft and provides bump softening, the rest of it takes care of the bigger bumps.
The damper -- Ill call it a shock from now on, although bugs the hell out of me -- is there to control the movement of the spring. Ever rode in someone's old caddy, or olds, and the car rolls up and down after hitting a bump like it's on waves? That's because the springs are still doing there jobs, but the shocks are shot, and are not controlling the movement of the spring. Ideally, the spring should only compress/decompress once. The shock is what controls the movement of the spring.
Jesus, this is getting long. You better make this a sticky.
The shock has two "settings"...at least. There is the rate of compression, the rate at which the shock allows the spring to compress. YOu dont want the spring to just shoot up and end up bottoming out every time you run over a twig (exaggerated example). Too stiff, and the spring doesnt give at all, and the car goes bouncing up in the air.
THe second setting is the speed at which the shock moves the spring back down to the pavement. The stiffer the shock -- to a point -- the quicker the wheel gets back down to the ground.
Okay, got all that?
Coilovers are combination shock/spring combos, most often adjustable. Do a search on the site to find them. You will see a spring on an adjustable perch on the strut, and within that spring you will see the shock (spring damper).
When looking at aftermarket coilovers, you want to compare a) the spring rate with your current spring rate and b) the compression and rebound stiffness of the shock. Too soft, and you get a smoother ride with decreased handling and increased body roll, too stiff and you're riding in a kart down city streets.
I didnt get into how the shocks physiologically work, through hydraulics and heat. Someone else can do that.
Hope this helps.