Radiance is the physical quantity related to light intensity: the power of the light, spreading out in some solid angle, over some area.
Luminance is a photometric quantity which is basically the effect of radiance on our eyes. Its a single scalar: the integration of radiance, weighted with this curve, which describes how efficiently different wavelengths of light can trigger the cone receptors in our eyes. Because luminance is a weighted integral of radiance, there is a nice linear relationship between the two.
Brightness, on the other hand, is the subjective visual experience of luminance. Roughly, it is the effect of luminance on the brain. Because its a subjective quantity, the relationship to luminance is non-linear, approximately a cube root.
Its important to understand this difference between luminance and brightness, because I'll be using the two terms throughout this talk.