I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold.
I read that in super-spaces you have 2 Grassmann numbers for each coordinate.
Could anyone explain to me what these Grassmann numbers are?
And then, what's the difference between a regular manifold and a supermanifold?
Supernumbers and their weirdness are e.g. discussed in this Phys.SE post.
The next logical step is to learn the notion of $(n|m)$ super vector spaces, which have $n$ Grassmann-even and $m$ Grassmann-odd dimensions.
Moreover, we will assume that the reader are familiar the definition of an ordinary $n$-dimensional $C^{\infty}$-manifold, which is covered by an atlas of local coordinate charts $U\subseteq \mathbb{R}^n$.
Finally let's discuss $(n|m)$ supermanifolds, which is technically a sheaf of $(n|m)$ super vector spaces. Heuristically and oversimplified, a supermanifold is a generalization of the notion of a manifold (3) where the local coordinate charts now are subsets of $(n|m)$ super vector spaces.
References:
Bryce DeWitt, Supermanifolds, Cambridge Univ. Press, 1992.
Pierre Deligne and John W. Morgan, Notes on Supersymmetry (following Joseph Bernstein). In Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, American Mathematical Society (1999) 41–97.
V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, 2004.
There is a very detailed discussion of supernumbers, supermanifolds, and other superstuff in DeWitt "The Global Approach to Quantum Field Theory". If you are more mathematically-minded, look at this book on mathematics of QFT and String Theory.
Here are detailed online lecture notes that introduce Grassmann coordinates, supergeoemtry, supermanifolds etc.:
