As the name suggests, shape is nothing more than the intrinsic shape of the diamond.  As viewed from the top, the most common shapes are:

Round	Princess	Emerald	Radiant	Oval	Pear	Marquise	Heart

With the exception of the emerald cut, today's most popular shapes are brilliant cuts or hybrid cuts (as opposed to strict step cuts).  In terms of cost, the round brilliant tops the list, while the square cuts typically cost about 10% less for a diamond of the same weight and quality. 

Unless you're purchasing a diamond as an investment (in which case you should strongly consider a round brilliant), you should choose the shape which pleases you most.