What is meant by "set map" or "map of sets" in this context. I am reading about free $R$-modules. There is a line that says "For any set $A$ there is a free $R$-module $F(A)$ on the set $A$ that satisfies the following: if $M$ is any $R-$module and $\phi:A\rightarrow M$ is any map of sets...."
What is meant by 'map of sets' in this context? What would the collection of all such maps be?
Is it just saying the function is from $A$ to $M$ and we can mod it in the obvious way to be from the subsets of $A$ to the subsets of $M$?

A "map of sets" just means a map that doesn't necessarily bother to preserve any structure. It's best explained by contrast: a "map of groups" is a map that preserves the group operation. A "map of rings" is a map that preserves both ring operations. A "map of sets", by contrast, is a map that need not preserve anything at all.
Another way of saying it might be to say "a map of just sets" or "just a map of sets". They're saying it like this to make it explicit to the reader that they are not talking about a group homomorphism, or a module homomorphism, or a ring homomorphism, or any other kind of map you've encountered recently; they're just talking about a plain old vanilla map.
