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


The definition of subset;
The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$


we know what a subset is and what a power set contains.

In a simple case where a set $A$ is supposed to exist, with $A=\{a, b, c\}$, we know what is and what is not a subset of $A$: 

$\{a\}, \{b\}, \{c\}, \{a, b\}, \{a,c\}, \emptyset$ and $A$ are subsets of $A$ and anything different is not.


  $\wp(A)=\big\{\{a\}, \{b\}, \{c\}, \{a, b\}, \{b,c\}, \{a,c\}, \{a, b, c\}, \emptyset\big\}$. 


However the mere definition of something (and consequently it's recognition as such) does not guarantee its existence. $\emptyset$ and $A$ seem like the only subsets whose existence is immediate.

In other words, I know what a power set contains, but how do I know that the things it contains exist in the first place?

Because such a well-defined and existent set such as $\wp(A)$ should not contain nonexistent elements, to prove the existence of its elements is important. It seems that two alternatives arise:


Being a member of $\wp(A)$ automatically makes this thing to exist;


or


There should be an alternative to prove the existence of all subsets of $A$ without the axiom of power 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 $|\mathcal{P}(A)|&gt;|A|$, 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”.
The most relevant axiom is actually the one mentioned by MathematicsStudent1122 and MauroALLEGRANZA, namely the Axiom of Separation (sometimes called specification).  Whenever you have a property expressed for example by a formula, this axiom guarantees that there is a subset whose members are precisely those entities that have the property or satisfy the formula.

For example, you can write down the property of a natural number $n$ that $(\exists m\in\mathbb{N})\,\big(n=2m\big)$ meaning that $n$ is an even integer.  But how do you know that the set of even integers exists? That's the role of the separation axiom.

In certain interesting mathematical theories, you can have predicates that do not satisfy separation. One such theory is Nelson's.