when we say that t A is an nxn matrix whose columns are given by the vectors in S where S is a basis for R^n,
does it mean that A is a transition matrix from S to S'' where S'' is the standard basis in R^n?
You are quite right.
To elaborate, if is the given basis for , and if is the matrix, whose th column is the basis vector , then if is the coordinates of the vector with respect to the (ordered) basis , then
and the vector can be considered as itself written in the standard basis.