The surface density of something is the amount of it per given area of a surface, often signified by Σ. For example, the mass per surface area or luminosity per surface area. In astrophysics, it may be used for flat objects, or may be used merely because it is the observable quantity, i.e., of an area over the celestial sphere, when the depth of something is less clear. As such, it may be used when volume density might seem to make sense, e.g., the Kennicutt-Schmidt law.
Measurements of qualities across resolved images of largely-flat things, such as disk galaxies and circumstellar disks, (e.g., luminosity, density of molecular clouds, etc.) reveal a surface density distorted by the distance and angle of view. Determining an actual surface density depends upon determining or accommodating these factors and also depends upon the transparency of the object to the quality of interest.
Surface density is equivalent to column density, but with some different connotation regarding the shape of what is being measured.