Why is there a band gap in semiconductors but no band gap in conductors? At this point, I understand that at least one fundamental difference between conductors and semiconductors is that in conductors, there is typically no band gap because the valence band and conductance band overlap one another; in semiconductors, there is no such overlap meaning there is an energy hurdle to be overcome before an electron can break free from the valence band and move about in the conduction band.
What I still don't get is WHY is that the case? Is this just a simple fact of electron configuration such that I should be asking the phys-chemists at Chemistry Stackexchange about this? Is there something going on at the fundamental particle level that accounts for the differences in behaviors? Do we even know the "why" of this yet?
Also, this may be related: as you pare down the size of a sample of conductive material and get into the range of perhaps a couple dozen atoms, you achieve quantum confinement which means that you've created a band gap between valence and conductance bands. So what property of matter is it that accounts for this?
This is all a single question, I'm just trying to approach it from a few different angles to underscore my confusion. Thanks for your input!
 A: If you just take the empty bandstructure, you will see that any periodic arrangement of atoms (conductors, semiconductors, insulators) features a set of allowed bands and forbidden regions, so called bandgaps. Fully occupied bands can not contribute to electrical current. There are no free places, where carriers could move. Only partially occupied levels allow current.
The difference between these groups of materials is, where the Fermi level lies.
In the case of a conductor, the Fermi level is in one of the bands, therefore at least one of the bands is partially filled and can carry current.
In the case of a semiconductor, the Fermi energy lies within the bandgap. In an ideal case of a defect and contamination-free (undoped) semiconductor, it would be in the center of the bandgap. Carriers can thermally be excited and lead to some intrinsic conductivity as there are some electrons in the conduction band, which naturally leave holes in the valence band. Both can carry current. The amount of free carriers can also be engineered through doping with impurities (e.g. P or B in Si, Ge or Si in GaAs).
Insulators are basically semiconductors, where the bandgap is so large that only a negligible amount of carriers is excited thermally.
As indicated by the comments to your question, there are many special cases with zero bandgap, negative bandgaps, overlapping bands, ...
What is the "origin" of these bands?
You start from Schrödinger's equation with a periodic potential and apply Bloch's theorem to solve with a periodic wavefunction. I found a good explanation at physicspages.com, but there are also several articles in wikipedia, which explain this. Not to forget, this is treated in about any solid-state physics textbook.
