There are two stages of sharpening plus there is also usage of sharpening tools for other creative purposes.
1. Input sharpening.
Digital sensors are actually an array of cells which sample the light impinging on them (lets for a moment neglect bayer and other color formation related issues). In front of the actual sensor cell there sits a number of layers, some of which are part of the sensor itself. Some of these are built into the chip, such as the color filters and oxide, and passivation layers deposited on the silicon as part of sensor manufacturing. Others are assembled as part of the unit installed in the camera, perhaps a low-pass (AA) filter, and maybe a protective glass sheet. All of these impart blur to a greater or lesser degree.
Lenses also impart blur and veiling flare. Not only glass imperfections, but internal reflections, not to mention reflections that come off the sensor, hit the rear element or some other surface where a refractive index change might occur, and are returned to the sensor.
De-bayering and noise reduction also impart blur and color smearing.
The purpose if input sharpening is to reduce the visible effects of all the above.
One component is to increase the pixel-to-pixel contrast One way to do this is through the use of unsharp mask with a low radius setting (usually less than a pixel). The other component is to combat veiling. Sometimes this is done with unsharp mask with a low percentage large radius setting. The "clarity" slider of many converters is often implemented this way.
Of course there are other techniques (such as deconvolution LOL), but these steps are all designed to produce an image that is apparently free of these blurring and veiling artifacts.
2. Output sharpening
Each step in the image reproduction process introduces some sort of edge blurring effect. Inkjet printers effectively paint the image on paper with nano-sprayguns. Printing processes, no matter what technology used so far do their bit to fuzz things up. So the purpose of output sharpening is to attempt to compensate. Output sharpening is dependent on issues such as the reproduction technology, image size, and viewing distance.
Since each time an image is printed it needs to be sharpened appropriately for the specific purpose, it is left to the end as well as other compensatory adjustments such as curves and white-point and black-point compensation.
Separating sharpening into two stages enables one to standardize the process to some degree. Each camera and lens combination (and sometimes shooting situation) might need different treatment, whereas each printer/paper/size/distance combination another. If the steps are combined, then one is faced with an experimental trial and error situation which may waste both time and materials.
-bob