supermanifolds - elementary introduction?

There is a short elementary survey by Stolz and Teichner: http://www.boma.mpim-bonn.mpg.de/data/28screen.pdf

Some further references, that might be of interest for your purposes:

• You can see at this article and the book Supermanifolds Theory and Applications by Alice Rogers. The article discusses -among others- the relation between the De Witt approach to supermanifolds and the approaches and definitions of Kostant and Leites (see also lower in this list). In the book, several topics of interest in physics, such as for example super Lie groups, the super Poincare group, Grassman algebras, $N=1$ supersymmetry, supergravity, topics and applications from string theory etc are studied from a modern algebraic-geometric point of view.
• The book Supermanifolds by Bryce DeWitt.
• The article Introduction to the theory of supermanifolds, D A Leites 1980 Russ. Math. Surv. 35, 1. (this has also been mentioned in the comments aboven and is cited here for the sake of completeness).
• The conference paper Graded Manifolds, Graded Lie Theory, and Prequantization, B. Kostant, Lect.Notes Math. 570 (1977) 177-306, in "Bonn 1975, Proceedings, Differential Geometrical Methods In Mathematical Physics", Berlin 1977. This paper states the definitions and basic notions in a more general setting (see also the work of Leites mentioned above) -at least in my understanding- but contains an extreme wealth of information and lots of detailed proofs which are not easy to be found in other sources.
• The article The structure of supermanifolds, Marjorie Batchelor, Trans. Amer. Math. Soc. 253 (1979), 329-338 (also cited as ref [1] in the reference provided in the answer by user Dmitri Pavlov).

Finally, it is interesting to mention the Wikipedia pages on Supermanifolds and Graded manifolds which attempt to discuss the relations between the various definitions met in the literature.

Some further (further) references:

• Lectures on Supergeometry by G. Sardanashvily;
• Lectures on Supergeometry by Tiffany Covolo and Norbert Poncin;
• The notes by Witten: 1 and 2;
• Supersymmetry for Mathematicians: An Introduction by Varadarajan (in particular chapter 3 and 4)
• Supergeometry, Super Riemann Surfaces and the Superconformal Action Functional, by Enno Keßler.
• Notes on Supersymmetry, by Pierre Deligne and John W. Morgan (following Joseph Bernstein), which are the first four chapters of Quantum Fields and Strings: A Course for Mathematicians;

(Don't be deceived by the title of the last reference: the first four chapters are about superalgebra and supergeometry, not about SUSY.)