What are the essential elements that make up a painting? In no particular order, composition, drawing, perspective, values, edges, colour – all of these contribute to the outcome. Edges are arguably the least appreciated of these, but contribute greatly to the three-dimensionality of a painting that aims at being representational.
I was re-reading Charles Sovek’s Catching Light in your Paintings recently, in which he lists the four kinds of edges – Hard, Firm, Soft, and Lost.  Hard, and Lost, are probably the easiest on figure out.
Hard edges are clear, well-lit, in focus, and catch the eye.
Lost edges are valuable in all styles for taking bits of the picture out of focus, in contrast  with the bits that are sharp and meant to take the attention. Lost edges automatically occur when two edges share the same dark tone, even if the colours are different. I find them especially useful for getting edges that are on the shadow side lost in the dark background that l often use.
Soft edges occur when there is a transition of colour or value (tone) with no distinct margin – you can simply soften adjacent edges by blurring them or blending them together.
Firm edges are somewhere between hard and soft -  a distinct transition, but clearly neither sharp nor blurred.
The image below illustrates the different types of edge. In the pastel with the Samovar and Tilly lamp, there are two light sources – the set-up is lit by a spot from the right, and the Tilley lamp has an internal light. 
 
The lemon is the brightest object, and as you go around the circumference you proceed from a Hard edge to a Soft edge – and internally there is a Soft edge between the half-tone and shadow. The reflection of the lemon goes from Firm to Lost. The samovar goes from Hard where it is highlighted, to Lost in the shadow, along its perimeter. Internally there are firm to soft passages. Likewise for the Tilley lamp, specially the top.