Introducing: Your rear suspension
To help you better understand what happens when your rear wheel moves up and down, we present two illustrations. The animation on the left shows how everything back there fits and works together. The one on the right is for those who grew up before MTV.
This view is from the left side of the bike, so the front is at the left side of the picture and the rear at the right.
Explanation of suspension damping
This deals mostly with how the rear suspension acts to keep your wheel on the ground, but the front works the same way.
Damping is the shock's ability to prevent the forces of the road imperfections from being repeatedly transmitted to you through the bike's chassis. Holding the bike's chassis from the tires are springs. Their job is to keep the bike suspended in the air, at a set distance from the surrounding parts.
When the tire hits a road imperfection it moves up and down. The spring compresses and stretches in order to prevent the chassis from moving up and down in relation to the imperfection. A spring, on its own, will keep expanding and compressing over and over again. So, to stop the movement of the spring, you have a shock absorber.
Now, if you have a shock with a high amount of damping, then very little of the spring's movement gets transmitted to the chassis. This makes for a nice smooth ride, but it feels like you are riding on a marshmallow. If you have very little damping, then most of the forces do get transmitted to the chassis, and it feels like you are riding on rocks. So, the adjustments that are on adjustable shocks are there to allow you to bring the suspension to a happy medium between riding on a marshmallow and riding on a rock.
Set your (adjustable, non-stock) shock to what feels best to you under all conditions, and you will be happiest with the bike.