Given any set$S$, a family of subsets of $S$ consists of an index set $I$ and and a function from $I$ to the power set of $S$. This is actually nothing special; this is the normal meaning of ‘family’ when taking a family of things that form a set: a family of foos is any function to the set of foos.

However, in predicative mathematics, we do not accept that the subsets of $S$ form a set; that is, we believe that the power set $\mathcal{P}S$ is really a proper class. So on the face of it, a family of subsets would also be a large thing. However, there is an alternative definition that is not only appropriate predicatively but also often provides a useful perspective in any case: A family of subsets of $S$ consists of an index set $I$ and a binary relation$\in$ between elements of $S$ and elements of $I$.

Note that we may have $x \in A \Longleftrightarrow x \in B$ for all $x$ without having $A = B$; this is because (as in any family) two different indices may index the same subset. However, many of the operations performed on families of subsets, such as intersection and union, depend only on the image of the function $I \to \mathcal{P}S$. So it is common to assume that a family of subsets of $S$ is simply a subset of $\mathcal{P}S$. (Besides this, not all authors agree with this page on even the general meaning of the word ‘family’.)