Most sensors in digital cameras use an anti-aliasing filter.
http://en.wikipedia.org/wiki/Anti-aliasing
One result of the AA filter is an obvious softening of the image. Images out of the camera in JPG format, most noticeably from the non-DSLR type, apply sharpening algorithms to improve the image by contrast sharpening. Images that have not been sharpened in the camera generally require some sharpening in an image editor, the degree of which is dependant on the sensor + AA filter, type of image, e.g. portrait vs. landscape, and the lens used.
The automatic white balance in most cameras can be poor, particularly when the scene/subject is illuminated with mixed light sources of different temperatures, e.g. fluorescent & tungsten lights. This may give an overall yellow, green, etc. cast to the image and may require correction in post processing.
Based on the type of sensor and its output, some images may ‘improve’ with a little adjustment to saturation and contrast as well.
The general rule, as several folks have posted, is to get the shot right from the beginning. And then there may be a need for adjustment in post-processing, particularly sharpening. However if the composition, exposure, etc. are correct from the onset there is minimal post processing required. I think all P&S do in camera processing, and more DSLRs include this with ‘shooting styles’.
Shooting in RAW format produces images with essentially no in-camera processing or compression, and these will require post processing. There are a few advantages to shooting in RAW, but this format may not be necessary and will depend on one’s shooting style and needs.
There are techniques such as High Dynamic Range (HDR) in which the photog wants to produce an image that exceeds the dynamic range of the camera. This may require multiple shots with different exposure settings and merging the shots in PP. High Key and Low Key are other techniques that also depend on PP.
It may be interesting to note that Ansel Adams, in addition to being a superb photographer, was a master in PP but it was darkroom based - both optical & chemical. Drop a roll of film off at any developer and the technician or computer will adjust the print parameters to suit the image needs. It’s still post processing.