I remember that I was going to give essentially the same answer to this question as Hartmann but apparently I forgot! :) However, it can be given in a more conceptual way (using stacks or groupoids) generalizing the above beyond curves and germs (see the reference below).
Have a look in one of the appendices to the book "Introduction to Singularitites and Deformations" by Greul, Lossen and Shustin (Springer-Verlag). The book is however mainly concerned with "complex space germs" and their deformations but the appendices treats some more general stuff. In particular, there are references to more general referencestreatises.
If you want to go really hi-tech, try the paper "Versal deformations and algebraic stacks" by Michael Artin (hard to read though).
