The CLEAN (Högbom 1974) algorithm represents the image as a number of point sources in an otherwise empty field of view. A simple iterative procedure is used to find the locations and strengths of these point sources.
The Högbom CLEAN algorithm looks for the brightest (absolute sense) pixel (called a CLEAN component) in a specified region in the image (called the CLEAN window which must include all the real emission in the image) and subtracts a fraction (called the loop gain, and generally about 0.1 or less) of the dirty beam from the dirty image at the location of that brightest pixel. This subtracted image is called the residual image. The search and subtraction loop is repeated until the sidelobes in the image are reduced to below the noise level. It is easy to see how CLEAN works if you just think about a single point source. For extended sources, one thinks of the emission as a collection of point sources.
After CLEANing is finished, the model of the source that you have constructed consists of a list of CLEAN components at various locations in the image. Looking at the CLEAN component image is a sobering experience (you can't do this easily with AIPS though). To improve the qualitative appearance of this model, and to suppress what is essentially unmeasured high spatial frequency structure (i.e., structure smaller than the resolution), CLEAN generally finishes its work by convolving the CLEAN component image by a Gaussian. This convolved image is then added to the residual image. The Gaussian is chosen to match the main lobe of the dirty beam, and is called the CLEAN beam. The AIPS task HBCLN performs a Högbom CLEAN.
I mentioned that the units of the dirty image are not very useful (§ 15). However, convolved CLEAN components do have meaningful units of Jy per CLEAN beam area, which can be converted to Jy per unit area, because the equivalent area of the CLEAN beam can be calculated.