I always thought that pressing takes place the moment that grapes arrive to a winery. This is the case with white wine, because you want to separate the juice from the skins immediately in order to limit the amount of color added to the juice. With red wine, you want all that color. Further, most of a wine's character comes from the skins.
As I mentioned a few posts ago, juice and skins are often left in contact for 15-30 days. An act of artistic expression, some winemakers choose to immediately press the skins when the process of fermentation is complete. As Charles Scicolone writes (in a particularly revealing post about Barolo wine), other winemakers choose to leave the skins in the juice after fermentation has finished. These latter wines can exhibit a large number of possible differences, in particular, they become more concentrated.
Below is a slideshow that shows the gushing process of pressing. The grape skins that are pressed are those left at the bottom of the stainless steel tanks after the fermented grape juice (now wine) has been transferred, or racked, in another tank. Pressing is another artistic process because it determines how tannic a wine will be. The more times that the skins are pressed, the more of their intrinsic flavors are released, including tannins and tartaric acid.