A zero thickness error can occur any time there is solid geometry that touches itself at a point or an edge. For example, if the area between the large rectangle and the small rectangle in the image below were extruded to form a solid, the point where the small rectangle touches the large rectangle would form a line of zero thickness.
Zero thickness is a problem for a couple of reasons. First, it is impossible to manufacture in a single piece. Second, it causes a lot of other problems for the solid modeling process. Third, it is probably almost always a mistake, since there are no truly practical applications for this sort of geometry. It is essentially knife edge up against knife edge.
This may be the kind of thing that you would create from two different pieces of material, but not with a single piece.
This is a topic that has seen a lot of heated debate, and the CAD companies (Solid Edge included) tend to take the view that zero thickness is an error that should be avoided, while some users argue that there are some isolated situations where you really need that particular tool.
There are only a couple of situations where the argument against zero thickness as an error have any real validity. One is the case of a helical spring in its compressed stack state. In this situation, you have line-to-line contact, and it is manufacturable.
Another situation would be sheet metal where a flange is folded over flat to the part. This gives face to face contact with zero thicnkness gap. This is also obviously valid from modeling and manufacturing points of view.
In both cases, the workaround is the same: a very slight gap or overlap. If you take the measurement of "zero gap" down small enough, say to the molecular level, there will always be a non-zero gap between faces just due to molecular forces.
So when you see an error such as the one shown above or you hear someone tell you you have a "zero thickness" error, this is the kind of thing they are talking about. Zero thickness situations within a single body are generally not allowed in 3D CAD, so your workaround is to create a small gap or a small overlap, with the small gap being preferred