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Intensity in astronomy (commonly called radiance outside astronomy) 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. This particular measure has the property that it does not decline with distance, defined in such a way that it remains unaffected by the spread of EMR.
The general strategy of this measurement is to quantify the energy of the EMR traveling from a finite-size portion of the source to a finite-size portion of the recipient, then taking the limit of this as the two portion-sizes are reduced to zero. More accurately, it is a such a limit on the EMR energy 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 covers a specific angular diameter of the sky (from your viewpoint), then the EMR from that portion of the Sun striking a specific plot of ground has a particular intensity, and (somewhat non-intuitively) if the Sun were closer, or somewhat further, the intensity from that angular diameter would be unchanged. Along a mathematical line (of infinitesimal width), the EMR energy must be zero, and intensity 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 intensity of two sources can be reasonably compared as one being (for example) twice the other.
A common unit of intensity is watt per steradian per square meter.
∂²Φ
L = ————————
∂A∂Ωcosθ
Φ
≈ ———————
AΩcosθ
Specific intensity or spectral radiance is the intensity at a specific wavelength. The mean intensity is the average intensity in all directions from a surface (perhaps within a solid angle), i.e., integrating it over the angle and dividing by 4π. The specific mean intensity aka mean specific intensity is this per wavelength. Mean intensities can be useful in simplifying models.
In more general physics, the term intensity is commonly used with a different meaning, i.e., all electromagnetic radiation striking a surface in watts per square meter. Thus the general physics term radiance for the above-described meaning.