Supergeometry in mathematics and physics by Kapranov (arXiv, 34 pages, submitted on 22 Dec 2015, last revised 2 Apr 2018). Abstract:

This is a chapter for a planned collective volume entitled "New spaces in mathematics and physics" (M. Anel, G. Catren Eds.). The first part contains a short formal exposition of supergeometry as it is understood by mathematicians. The second part discusses aspects of supergeometry that are used by physicists in relation to supersymmetry. Finally, the third part is an attempt to uncover homotopy-theoretic roots of the super formalism.

On the afterthought, I decided to add here earlier work by Manin - since Kapranov acknowledges his guidance, but also since it is still quite informative (I think), and has not been mentioned here so far.

Chapters 3 - 5 of his "Gauge Field Theory and Complex Geometry", as well as Chapter B of the appendix to the second (1997) edition by Merkulov provide a self-contained exposition of superalgebra and supergeometry, with a description of physical applications.

"Topics in Noncommutative Geometry", especially chapters 2 and 3, provide, I believe, sort of a continuation of the above.