Sharpness is the opposite of softness but there is a whole spectrum between them. In a soft image, details are less pronounced which means that the difference between adjacent pixels is smaller. Conversely, when an image is sharper there is more contrast between pixels which is sometimes referred to as micro-contrast as opposed to overall image contrast.

The level of sharpness depends on the entire system, not just the lens but also the anti-alias filter all the way to the method used to interpolate colors from sensors which use a color-filter array (Non-Foveon sensors basically).

Check out this demo from the Pentax K-5 review. The K-5 and K-7 have the most sophisticated sharpness control of any digital camera. Sharpness is controlled in 9 steps on one of 3 scales. The default position is 0 but if you select -4 at the bottom scale you will see a very soft image. If you select +4 you will see a sharp image. Note that the setting of +4 is what people call over-sharpened which means the contrast between adjacent pixels was increased so much that artifacts are introduced (just look at the edge of the parenthesis on the left).

You can also select another scale and see the difference between the Sharpness and the Fine Sharpness or even Extra Fine Sharpness scales. These use different sharpening parameters which are designed to work on different levels of details. At the extreme of the Extra Fine Sharpness scale, it even sharpens noise!

What is really interesting is that this is the same image which you can see with different levels of sharpness. Only one shot was taken and it was developped in-camera from DNG to JPEG uses all possible sharpness settings. Of course, the sharpness you start with and how much you get out of it depends on the quality of the lens used. Some lenses inherently produce softer results which you can sharpen by software but limits to how far you can go without introducing nasty artifacts. Even worse is that the sharpness of a lens is not uniform so will usually by highest in the center and also changes with the selected aperture.