What is slow cooking and what it is good for?
Generally, slow-cooking means any food preparation method which relies on using low-heat for a long amount of time. Barbecues, smokers, luau pits, and low-heat ovens could all qualify.  The benefit of slow-cooking, generally, is that food becomes incredibly tender, as all of its connective tissues break down.  Also, flavor can infuse over time and provide deeper results than with virtually any other method.  Lastly, long-cooking times create a celebratory atmosphere where food is the focal point of a social gathering.

Specifically, slow-cooking refers to the 'slow-cooker' a counter-top set-up which has a large, electronically heated ceramic bowl and a glass lid.  It usually has only a few settings, such as 'low', 'medium', and 'high'.  The benefit of slow-cooking in a slow-cooker is mainly convenience.  These units are safe enough to turn on before work and return to before dinner, with fully-cooked, piping hot food ready to go.  The slow-cooker is geared towards 'one-pot' recipes, which means they require minimal food preparation.  You simply cut up the ingredients in fairly large chunks, add seasoning, and liquid, and that's it. They are a favorite of working parents, but increasingly, also a device for more advanced experimentation.