What are light echoes?

A light echo is reflected light from a light source. Similar to the muffled echo of sound when someone shouts something towards a wall, the light echo still contains the initial signature of the original light, even if somewhat rendered. The light echoes we are talking about are reflections of Supernove light off interstellar dust. These are true reflections, i.e.  the original spectral distribution is largely conserved. It is NOT absorption and then re-radiation of light. The graphic below shows a schematic of the geometry of the light path that creates a supernova light echo, shown as if the process could be viewed from the side.

The Earth is in one focal point, and the Supernova is at the other focal point of an imaginary ellipse. A property of an ellipse is that all paths from one focal point to any point of the ellipse to the other focal point have the same length. Thus any light coming from the Supernova reflected off dust that intersects the same ellipse reaches the Earth at the same time. Thus for any given time after the explosion there is a corresponding ellipse, and we only see a light echo at that time if there is dust intersecting this ellipse.

In three dimensions, there is an ellipsoid. since in general the dust is in sheets or filaments, the cross section of these dust sheets has the shape of circles or arclets of circles, depending on the size of the dust sheet compared to the ellipsoid. See below for an artist conception of light echoes.