The terms hard edge and soft edge are used to describe two different ways in which objects can be painted. A hard edge is the term used when the edge of an object is painted in a well defined or definite way. There's a strong sense of where the object ends. A soft edge is when it is painted so that it disappears or fades into the background.
Take a look at this lily painting by Monet and compare the edges of the various lily leaves. Notice how some are clearly defined (hard edges) and some (especially towards the back on the right-hand side) dissolve into the blue of the water (soft edges). Your brain still interprets both as lily leaves even though they aren't all painted the same way.
