What are polyhedrons? Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ to be the number of joints where two faces come together side by side $n-e+f=2$.
It was later seen that a serious defect in this definition (and in the proof supplied by Euler) is that it is not at all clear what is a polyhedron in the first place. For example if we consider a cube nested within another cube as a polyhedron then $n-e+f=4$, a counter example to Euler's result.
What will be the modern definition of that polyhedron which will comply with Euler's result?

The polyhedra that Euler studied was not the same as we call polyhedra today. Today a polyhedra is a body with (only) flat polygonal faces.
Those that Euler studied was a subset of these. Basically those whose nodes and vertices form a planar graph.
There are some complications that can arise in the generic definition that Euler did not consider. 
The most striking is that it that it allows for cutting out the inside of it, that is that it's surface would not need to be connected. 
Another is that you may have donut-like polyhedra, ie take a polyhedron and make a prism shaped hole through it.
A third is the assumption that the edges and vertices might not form a connected graph. For example if you glue together two differently sized cubes at one face (so that the smaller sits on the face of the other).
The second construct will decrease the euler characteristics and the third will increase it. The consequence is that you could combine these complications to result in a polyhedron that Euler didn't consider, but that nevertheless satisfies his formula.
