In mathematics, a manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. ... 

shortest description i've heard that i thought helped was that a manifold surface is one in which every edge is adjacent to two faces. i believe that property is necessary and sufficient.

e.g. a triangle is not a manifold surface - each edge is adjacent to a single face. a cube is a manifold surface. if you take one face away from the cube, it is no longer manifold.

some properties that come out of this - no open boundaries (like the cube missing a face), the surface is "air-tight", it has a distinct inside and outside surface, etc.

non-manifold is any surface that doesn't meet the requirement for a manifold surface.