Given a matrix $A \in \mathbb{F}^{x \times y}$ and a set $C \subseteq \{ 1, \ldots, y \}$, what does $A_C$ mean? I have seen this notation in some places, but I am still not sure what it means.
It seems like this could be the extraction of specific columns from the matrix. If that is true, how would I differ between row and column extraction?
I think you are correct in your interpretation. Some possibilities for denoting row and column extraction would be $A[S_{1},S_{2}],$ $S_{1}\subseteq\{1,\ldots,x\},S_{2}\subseteq\{1,\ldots,y\},$ or $A_{S_{1},S_{2}},$ but I think that the best approach in any case is to clearly define what is meant by your choice of notation, because (as far as I'm aware) there is no standard way of denoting this.
In their excellent book Matrix Analysis, Roger Horn and Charles Johnson use the notation $A[S_{1},S_{2}]$, which is perhaps an argument in favor of this one.