Question

how is the resistance and the brightness of the light bulb related to the current through it?

Answer 1

Generally the bulb consists of a filament made up of high melting point and the having the high resistance value

as the current passing through the filament then having the resistance the electrical energy lost , or the energy will be dissipated at the resistor which in turns in the form of heat energy finally comes out as light energy corresponding to the radiation of that heat energy from the heated filament.

the power dissipation at the resistor is P = I^2*R

if we increase the current the dissipation energy also increases there by the brightness also increases .