Surface Brightness is, in effect, the brightness "per area" of an object, i.e., brightness per unit area. Surface brightness has the seeming-paradoxical quality that, assuming the object is sufficiently close to be resolved, the observed surface-brightness does not automatically diminish with distance; rather the apparent size of the surface becomes smaller. Thus, the surface brightness of the Sun as observed from Earth is the same as it is from much closer: it is merely that the Sun fits in a smaller "area of the sky", i.e., a smaller Solid Angle (number of square degrees) of the Celestial Sphere. Being near the Sun would be like having the whole sky as bright as (but no brighter than) that little patch of the sky covered by the Sun.
Since the surface brightness of resolvable objects is not apparently reduced by distance (e.g., as would be Magnitude), it is a valuable observable: it can be directly determined at any distance, even without knowing the distance. As such, it can sometimes be used to assist in determining a distance estimate. Resolvable objects (i.e., for which surface brightness measurements are possible) include galaxies, Solar System planets and Moons, Nebulae, and the Cosmic Microwave Background.
Arakelian Catalog (Ark)
de Vaucouleurs' Profile
Gas Fraction Estimation
Low-Surface-Brightness Galaxy (LSB Galaxy)
Surface Brightness Fluctuation (SBF)