The mathematical descriptions of manifold and non-manifold geometry can be quite confusing.

In layman's terms manifold geometry is something that can be manufactured following a specific set of mathematical rules. While, non-manifold geometry refers to geometry that cannot be manufactured following these rules.

An even easier way to put it, is that manifold geometry can be unfolded into a flat polygonal object with all its normals facing in the same direction and all its faces connected by edges, while non-manifold geometry cannot.

There are three types of non-manifold geometry:

(1) T type
(2) Bow_tie type
(3) Flipped_normals type

Here is a short video describing the three types of non-manifold geometry and how to detect and correct them in maya.

https://www.screencast.com/t/MzkzMjA0M

I hope you find this helpful.

Here are the mathematical rules that defines manifold geometry:

-Every interior edge belongs to two faces.
-Every interior vertex is surrounded by one sequence of edges and faces.
-Faces only intersect each other in common edges and vertices
-There is a material on only one side of a face, and the material is on the same side of all faces.


Cheers,
Rick M.