Non-manifold geometry is not ideal geometry. This is impossible to print but will also cause all kinds of issues, plus the entire universe will explode if you try to unwrap this in Maya for example.
I would like to explain what is non-manifold geometry and how to avoid it.
Non-manifold geometry is geometry in which a single edge shares more than two faces or several surfaces connected to one vertice. Sometimes interior faces also happen like the one below. This can be a nightmare to deal with later.
Sometimes this can happen by accident doing manual re-topology for example, or when welding points.
These cubes also share one edge, and this is similarly no good.
Technically also a box with open sides could be considered non-manifold. However this doesn’t pose that much of a problem and is sometimes used in games with no problem
What is common for all of these objects is that they cannot exist in real world. Thus they would bring immediate issues to 3D printing especially. But generally it is better to avoid them because they will issues to UV unwrapping, subdivision and such. So if we are to make model for 3D printing, every flat surface extending outwards needs to have thickness. But this also looks much more believable for rendering unless we are dealing with a very specific circumstance such as polygons to be used for effects or such.
It is better to avoid non-manifold geometry in the first place, following good modeling practices that do not result in such. Having several of these in a scene can be pain to deal with afterwards.
If we need to create a polygon standing on a box for example for some reason, it should be not connected to any of the edges of the box. Also the two boxes should not be connected in this way. If we need to create two boxes like this and to be one geometry, we need to connect it in a way that could happen in real world.
This for example is a way to model in a way that results in manifold geometry. Notice how this also looks more realistic.
Maya does have automatic tool to fix such issues in geometry but this does not often, or rarely works. It can also cause very severe issues such as geometry going completely haywire. I have seen models that have became un-fixable due to this.
Please also see my other good-geometry best practices guide to a concave and convex polygons.