How do you determine which star, between two, is more luminous? For example, if star 1 is 4x bigger than star 2 but both are the exact same temperature how do you determine which is more luminous?
One way to determine the luminosity of a star is to first determine its distance, d.  Then one can use the formula, Luminosity = Flux * 4 * Pi * d$^2$.  Flux is related to the apparent brightness that is measured with a telescope (and CCD or photometer).  mag = -2.5 * Log(Flux) + c.  

If the star is relatively nearby, one can use parallax to get the distance.  If the star is in a binary system, one can solve for the distance from the period, orbital velocities and apparent size of the orbit.  If you know the velocities throughout one orbit, then one knows how far it has physically traveled.  Comparing this with the apparent size gives an approximate distance.  However, since one usually does not know the inclination, this causes uncertainty.  If it is an eclipsing binary, then you know that it is not very inclined and this works well.  On the other hand, if it is eclipsing, one can usually determine the radii of the binary pair and then get distances from Stefan Boltzmann Law.

Another method for estimating luminosity is to locate the star in the HR Diagram (Luminosity vs Color).   The color (difference between the magnitudes in two different filters) locates it horizontally, and stellar class locates it vertically in the diagram.  One needs a spectrum of the star to determine the class.

And we have one more trick up our sleeves.  If the star is a periodic variable such as a Cepheid or an RR Lyra, then there are relations between period of variability and luminosity.