Is light brightness subjective or can it be quantified? What is the relationship between brightness perception and physical properties of light, such as light intensity?
 A: Well this is a case of loose usage of words or terms that have a common everyday colloqial meaning, but also (maybe) have a very specific scientific meaning, that is different from the common meaning.
You have used two words in this loose fashion.    "Light"  and "Brightness".
So let's first discuss "light".
The Commission on Colorimetry of the Optical Society of America, defines "light" as:- The  Psychophysical Response of the Human eye, to Electromagnetic Radiation generally in the wavelength range of 400 to 800 nm.
Thus "light" is NOT the EM radiation itself; but the human eye response to that radiation.  For that reason, we do NOT measure "light" in the same units as EM radiation; Watts , Watts/m^2, Watts / Steradian etc.
Light is measured in Lumens rather than Watts, and for a normal or so called "photopic" eye response, the peak of the eye visibility, at around 550 nm wavelength is 680 Lumens per Watt.   At very low dark adapted human eye response, there is a standard "Scotopic" response curve which is shifted to different wavelengths at the peak.
So back to "brightness" which is generally discouraged for scientific work because of it being imprecise.
The preferred term is "Luminance", which is the photometric equivalent of "Radiance" used for EM radiation of any wavelength.
"Radiance" is  Watts per steradian, per square meter, and "Luminance" is lumens per steradian per square meter.
Watts per steradian is "Radiant Intensity" and Lumens per steradian is "Luminous intensity"   Both units imply a mathematical point source.   In practice, the error will be less than 1% when measuring the intensity of a fininte sized source, if the measuring distance is at least 10 times the source diameter, for a non point source.
So radiance can also be expressed as radiant intensity per square meter, and luminance as "Candela" per square meter, where "candela" is the photometric unit of luminous intensity.
Now scientists in their typical loose shop usage, will use the word brightness for all sorts of things; actual visible radiation sources, or infra-red or ultra violet sources, or even for the electron emission from a thermionic or other cathode for example.
In all of those cases, they are using the term brightness as a substitute for "Radiance" or "luminance" or similar units all of which have dimensions of flux per steradian per square meter.   So the "brightness" of a thermionic or field emitter, would be Amps per steradian per square meter.
In the case of photometric units for visible "light sources", Luminance is quite difficult to measure, because you have to isolate sharply, a specific surface area on the source, and then define a very specific solid angle for the radiation being admitted to the detector, and that requires some focusing optical system.
Luminous Intensity in candela, is much easier to measure, because you don't have to focus on a specific source area, but you do have to measure from at least 10 source diameters away, constrained by means of apertures, or the finite size of the measurement detector.
It's a complex measurement problem, and it isn't made any easier, by loose usage of the terms "brightness" instead of "luminance" or using the word "light" for either UV or infra-red  radiant energy, which aren't visible.   "light" is NOT the radiation; but the human eye response to it; so "light" by definition is visible; but only inside the human eye.   Strictly speaking, we get NO LIGHT from the sun; our eyes make it from the visible parts of the solar radiation.
But even after being warned we still use the word "brightness" in shop talk; but we know what we really mean is luminance.    You might sometimes see the word "Sterance" instead of "luminance"
And use Radiance, or radiant Intensity, or irradiance (incident Watts per square meter) or emittance (emitted Watts per square meter)   The photometric equivalents would be "illuminance" for incident lumens per square meter, or luminous emittance for emitted lumens per square meter.
A: We often use the term radiance to describe the brightness of an object.  Radiance is the amount of energy emitted (or reflected) from an object per unit area per unit solid angle per unit time.
To answer your question directly, radiance is how we objectively measure an object's brightness.  It is similar intensity but accounts for solid angle
As an example, the sun irradiates the earth's surface with approximately 1000 Watts/meter$^2$.  This is an intensity, also referred to as irradiance, which again does not account for solid angle.
If there is a Lambertian object on the earth's surface, it will reflect equal amounts of energy in all upward directions, a Solid angle of $2\pi$ (it won't reflect into the ground).  If the object has a reflectance of 10% the object's radiance will then be
$$ \text{Radiance}  = \frac{\text{Irradiance}\times \text{Reflectance}}{2\pi} = \frac{50}{\pi} \text{Watts } \text{meters}^{-2} \text{steradian}^{-1}$$
We can also talk about the Spectral Radiance which is the amount of energy emitted per unit area per unit solid angle per unit time per unit wavelength.  In other words, it takes into account different wavelengths of light, or different 'colors'.  
The spectral radiance is important to take into consideration when talking about objects seen by the human eye because the human eyes response to the brightness of an object depends on the wavelength or color of the object and its illumination.  The field of photometry is dedicated to the study of how the human eye perceives the brightness of objects.
These terms are different from Intensity in that they take into account the solid angle over which the energy is propagating away from the object.  Intensity does not take this into account but is instead simply the energy per unit area per unit time.
