I'm trying to texture just a standard cube but some of it's polygons remain untextured (black). Here is the video:
What I'm doing wrong? Thanks
You are using the Blender Internal (BI) rendering engine, which is not physically based and thus does not interact with light realistically (as it's physically-based counterpart Cycles does).
The polygons are not actually black, they simply have no light falling on them (BI doesn't have the capacity to compute correct environment or bounce lighting). Try adding more lamp objects to the scene to illuminate the different sides of the cube, or simply drag the current
Point lamp around.
Alternatively, it is recommended that you use Cycles as your main render engine, not only for the realistic benefits that it adds but because the BI engine will no longer be included (or supported) in the upcoming 2.8 release.