The GPU is basically applying things to a finished image. It's like applying a filter in Photoshop
Its kinda in the name too "Post" ... "processing".
For the player - Its the effects you see in games like:
Depth of Field
Motion Blur
HDR - High Dynamic Range Lighting - Lighting of environment changes according to were the camera view is (Inside/Outside)
Colour Correction and Clamping - to limit the range of colours to give a more realistic feel or washed out feel.
GPU intensive (Shaders) and can stress graphics cards a lot more at higher resolutions.
DiRT is a good example of a Post Proccesing intensive game which can strain even the newest cards at extreme resolutions.
