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The common meaning of the term radiance (called intensity in some areas of astrophysics) is essentially a measure of the electromagnetic radiation (EMR) traveling along a straight line from some source to a recipient. It is commonly used in models, such as those of radiative transfer. Astronomically, it can only be measured for extended sources (e.g., for galaxies), often termed brightness. It is defined so as to have the property that it does not decline with distance, evading the effects of the spread of EMR.
This measurement of energy along a line is accomplished by considering the amount of EMR (i.e., its energy) traveling from a finite-size portion of the source to a finite-size portion of the recipient, then taking the limit of this if the two portion sizes are reduced to zero. More precisely, it is a such a limit on the amount of EMR striking a portion of a surface from a given solid angle (or equivalently, on that emitted from a portion of material over a given solid angle). If you define a small circular region of the Sun (as seen in the sky) that has a particular angular diameter, then the EMR from that portion of the Sun striking a specific plot of ground has a particular radiance, and (somewhat non-intuitively) if the Sun were closer, or somewhat further, the radiance for that angular diameter would be unchanged. Along a mathematical line (of infinitesimal width), the EMR energy must be zero, and radiance is merely the value of a distribution function, which yields actual amounts of EMR with appropriate integration to give it some width. But this resulting measurement can be treated mathematically, e.g., halved or doubled, and the radiance of two sources can be reasonably compared as one being (for example) twice the other.
The key to the mathematics is that the measure is defined in terms of an area at one end and a solid angle at the other. The mathematics (and the property that the measure remains unchanged with distance) is identical whether the EMR is traveling from area to solid angle or vice versa: light from a square arcsecond of the Sun (as seen from Earth) reaching a square centimeter of Earth is unchanged as long as the Sun is close enough that it spans the square arcsecond, in vice versa: light from a square centimeter of the Sun to a square arcsecond of the Earth (as seen from the Sun) is unchanged as long as Earth is close enough that it spans the square arcsecond.
A common unit of radiance is watt per steradian per square meter.
∂²Φ
R = ————————
∂A∂Ωcosθ
Φ
≈ ———————
AΩcosθ
The terms used for EMR measurement vary across the branches of science and technology (in some cases the same term has conflicting meanings) and in astronomy, term intensity is commonly used for this value, as described above, as well as the terms surface brightness and just brightness.
The terms spectral radiance, specific intensity and spectral brightness indicate the analogous values per frequency-interval or wavelength-interval, e.g., per hertz or per meter. Be warned that conversion between the per-wavelength and per-frequency spectral radiance is made more complicated by being based upon different differentials.