Are insulators and conductors arbitrary categories?
As you have expected, there is no sharp divide between the groups. The divide is man made.
Since all conductors have some resistance, (except superconductors - follow this link to find out more) and all insulators will conduct some current if they are forced to, this means there is no absolute dividing line between conductors and insulators. Since some metals are very, very good conductors with only a very small resistance, and some non-metals are very, very good insulators, the terms are convenient when we are dealing with the usual voltages encountered in a laboratory. (source)
In practice the definitions are helpful because the dynamic range between good conductors and good insulators is very large, and there are not very many situations where the middle ranges between conductor, semiconductor, and insulator prove useful.
Needless to say, those who work with high voltages (hundreds of kilovolts and above) will define the boundary between insulator and not-insulator very differently than a hobbyist tinkering with 5-10V. At higher voltages, we find the resistance of insulators starts to have more of an impact on our design, and we become more picky about how high that resistance needs to be before we call it an insulator.
Another way of distinguishing conductors and non-conductors or insulators is with band gap - for good conductors the fermi level of electrons is inside a band - semiconductors have a small band gap and good insulators have large band gaps...
Electrons in solids lie in energy bands, whereas in atoms and molecules they have generally sharp levels.
If you have a partially filled band then electrons can reorganize themselves within a band and easily allow a current to pass if a voltage is applied.
If bands are completely full no reorganization is possible, but above the highest filled band will be an empty band. - the filled is the valence band the empty one is the conduction band. If the gap between these is small then electrons from the valence can move to the conduction and then electrons can reorganize themselves a bit in both levels to allow current to flow.
If the band gap is large then it is hard to move electrons up and get a current to flow.