How can a light bulb run on different voltages (USA and EU)?
For a purely resistive light bulb (old style wire element) the difference between 110V and 220V on power is a factor of 4:

110^2/R * 4 = 220^2/R

The current will also change, not a good idea as 110V lightbulbs would probably burn out at 220V (the current would double, power would quadruple). 220V bulbs at 110V will be much more dim.

Other lightbulbs that are rated for both voltages (CFL's or LED's) might have circuitry that switches between voltages. 

Many other power supplies are rated for both and automatically switch between the two voltages. Some products have a selector switch between the two voltages. 

The load determines how much current flows.