Rembrandt, detail of self portrait from 1661
"Painterliness" refers to a quality of brush strokes, in which the brush strokes are highly visible, often looking thicker and tactile. It is contrasted with painting that is more linear, with a highly modeled, more controlled use of form, in which you don't see the brush strokes.
Johannes Vermeer, Girl with the Pearl Earring, 1665
Rembrandt's work, especially his later work, was painterly (see the roughness? the obvious brush strokes in this detail of a self portrait?); Vermeer's was not. In Vermeer's paintings it is often difficult to see the stroke of the brush at all.
Vincent Van Gogh, Portrait of Dr. Gachet, 1890
Van Gogh's work was painterly – the brush strokes here are even more obvious, and an important part of the painting as a whole. French academic painter William-Adolphe Bouguereau's work was not – his intention was to eliminate the brush strokes and give the appearance of what we would now call photo-realism.
William-Adolphe Bouguereau, Portrait of Gabrielle Cot, 1890
One method is not better than the other – they are simply different ways to approach the process of painting.
You can find out more about the origins of the term and the idea of painterliness here, at Wikipedia.