non-manifold geometry, going off the maya documentation definition, is "Non-manifold [geometry] polygons have a configuration that cannot be unfolded into a continuous flat piece [that is, when it is unwrapped for texturing]".
things like three faces sharing an edge, and suchlike.
In the case of edges, its usually where a single edge is just standing out in space, without an associated face (like the result from an 'Extrude Vertex' or similar).