How to determine the existence of all subsets of a set?

The power set axiom just tells you what it says: for every $A$, there exists a set $\mathcal{P}(A)$ such that $$ \text{for all $B$, $B\in\mathcal{P}(A)$ if and only if $B\subseteq A$} $$ There is no claim of “existence” of any particular subset of $A$. In $\mathsf{ZFC}$ one can show that $|\mathcal{P}(A)|>|A|$, so there is plenty of subsets.

It should be noted that, if $A$ is infinite, there is no hope to find, for each subset of $A$, a formula “describing it”, because $\mathcal{P}(A)$ is uncountable. This is however not a problem: the axiom tells you that you have a “container” for all subsets of $A$; when you prove that a set $B$ is a subset of $A$, then you know it belongs to $\mathcal{P}(A)$; and conversely, if you pick $B\in\mathcal{P}(A)$, you know $B\subseteq A$.

The real purpose of the axiom is that the subsets of a set form a set. In particular, for instance, the equivalence relations on a set form a set that can be isolated from $\mathcal{P}(A\times A)$ using a suitable predicate and the axiom of separation.

I remember some good notes about this in Paul J. Cohen's “Set theory and the continuum hypothesis”.