The notion of power object generalizes the notion of power set from the category Set to an arbitrary category with finite limits.
Let $C$ be a category with finite limits. A power object of an object $c \in C$ is
an object $\Omega^c$
a monomorphism $\in_c \hookrightarrow c \times \Omega^c$
such that
If $C$ may lack some finite limits, then we may weaken that condition as follows:
If $C$ has all pullbacks (but may lack products), then equip each of $\in_c$ and $r$ with a jointly monic pair of morphisms, one to $c$ and one to $\Omega^c$ or $d$, in place of the single monomorphism to the product of these targets; $r$ must then be the joint pullback
If $C$ may lack some pullbacks, then we simply require that the pullback that $r$ is to equal must exist. But arguably we should require, if $\Omega^c$ is to be a power object, that this pullback exists for any given map $\chi: d \to \Omega^c$.
If $1$ is a terminal object, then $\Omega^1$ is precisely a subobject classifier.
A category with finite limits and power objects for all objects is precisely a topos. The power object $P A$ of any object $A$ in the topos is the exponential object $P A = \Omega^A$ into the subobject classifier.
See Trimble on ETCS I for the axiom of power sets in the elementary theory of the category of sets.