What does being diffeomorphic mean in the context of configuration spaces? 
A sphere space can serve as a "model space" for any configuration
  space that is diffeomorphic to the sphere space.

This is a quote from my text book (Principles of Robot Motion: Theory, Algorithms, and Implementation, Howie M. Chose). What does diffeomophic mean in this context?
 A: A diffeomorphism is a smooth homeomorphism. (A homeomorphism is, informally, a function $X\to Y$ whose existence implies that $X$ and $Y$ are topologically "the same".)
So if we have a diffeomorphism $X\to Y$ then we know $X$ and $Y$ are "the same" in the homeomorphic sense. Many homeomorphisms can be smoothed out into diffeomorphisms, but if we have a very pathological homeomorphism then it may not be possible to give any diffeomorphism. This phenomenon, I am pretty sure, never arises for $3$-manifolds, but $\Bbb R^4$ has uncountably many structures which are homeomorphic but not diffeomorphic.
In any case, diffeomorphism of $X$ and $Y$ is therefore a stronger version of "the same"-ness, where we can expect the similarity to preserve not only topological properties, but differential properties as well. This point is a bit subtle, since no geometry must be preserved it is not necessarily true that analytic properties will carry over, but any topological results requiring differentiability will.
For instance, we would expect that any space diffeomorphic to the Euclidean $\Bbb R^4$ has an analogue of the mean value theorem, whereas a space only homeomorphic to this $\Bbb R^4$ might not.
