What can we learn from a band structure diagram? Other than the band gap and its magnitude, what are the things that we can immediately learn about the properties of the material just by glancing at its band structure?
Can we say something about the bonding? Can we see if the material exhibits some exotic properties like whether it is a topological insulator or not? Etc, etc..
 A: You can see, where the band maxima and monima are and thus see, which gap is direct or indirect. In the case of a direct semiconductor, the CB minimum would be straight above the VB maximum.
Bonding can not be seen directly, I think. But from the position of the Fermi level, you would see, if a material is behaving like a semiconductor or metallic. A bandstructure also implies a crystal structure (some periodic arrangement of atoms). I'm not sure, if this further implies at least partially covalent bonding.
A: Depending on what is shown in the band diagram, you can see if it's a topological insulator. Most band diagrams just show what's going on in the bulk material -- what you'd get with an infinitely large chunk of material with not surface. However, to be a topological insulator, you need surface states that conduct. These can be (and sometimes are) drawn on top of the bulk band structure. Since the surface states conduct, they have no band gap. See the first figure in the topological insulators wikipedia article.
As @engineer noted, you can also see if the material has a direct or indirect bad gap. The final thing I can think of is that by looking at the curvature of the bands near their extrema, you can get the effective mass of the electrons and holes.
I'm sure there are other uses as well.
