To understand how current flows in a material you first have to understand electrons behave in a material. The key feature of solid state physics is that many materials are crystals. This means that the atoms are spaced periodically. As you mention, band structures are the way that we summarize the effect of this periodic potential. Basically, a band structure just relates an electrons momentum p=mv=hbar k to its energy. The momentum can be positive or negative, the sign only denotes direction. In free space this is very boring, Energy=m v2 /2 = p2 /2m=hbar k2 /2m. When you throw in a periodic potential, this becomes modified and results in bands. Actually calculating band structures is quite difficult. The key idea is that there are ranges of energy where the electron can live and ranges of energy where the electron cannot live.