What the definition says is that given a functor $\mathcal F \colon \mathbf C \to \mathbf{Set}$ a universal element is an element $e$ of a $\mathcal F(r)$, for some $r \in \mathbf C$, such that every other element $x$ of another set $\mathcal F(c)$, for a $c \in \mathbf C$, it can be viewed as image of a unique function of type $\mathcal F(r) \to \mathcal F(c)$. This function must be of course the image of a morphism of $\mathbf C$ via the functor $\mathcal F$.
Universal element play an important role in mathematics, primarily because $\mathbf{Set}$-based functors are really important in maths.
First of all let's make a point clear: universal arrows and universal elements are essentially the same thing, I'll try to explain way.
The fundamental idea is that there's a bijection between a set $X$ and the set $\mathbf{Set}(\bullet,X)$, where $\bullet$ denote the singleton set $\{\emptyset\}$: we can identify every element $e \in X$ with the (unique) morphism $ \tilde e \colon \bullet \to X$ such that $\tilde e(\emptyset)=e$.
Let $\mathcal F \colon \mathbf C \to \mathbf{Set}$ be a functor, then a pair $\langle r \in \mathbf C, e \in \mathcal F(r) \rangle$ is universal element if and only if the pair $\langle r\in \mathbf C, \tilde e \colon \bullet \to \mathcal F(r) \rangle$ is a universal arrow from the object $\bullet$ to the functor $\mathcal F$.
Similarly given a functor $\mathcal F \colon \mathbf C \to \mathbf D$ a pair $\langle r \in \mathbf C, \tilde e \in \mathbf D(d, \mathcal F(r))\rangle$ is a universal arrow from an object $d \in \mathbf D$ to $\mathcal F$ if and only if it is also a universal element for the functor $\mathbf D(d,\mathcal F(-)) \colon \mathbf C \to \mathbf{Set}$.
Remind that categories are usually considered as object living inside the category $\mathbf {Set}$ so universal elements can be thought as a way to talk about universals of a category $\mathbf C$ in $\mathbf{Set}$-terms (externally to the category $\mathbf C$) while universal arrows are the way to think universals in the term of category $\mathbf C$ itself (internally to the category $\mathbf C$).
Both these point of view are actually useful: the first enables us to work in set-theoretic terms and so to work in a familiar framework, the second enable to work internally to the category we are considering, and this is really important from a logical point of view.
About the request for your application what are you looking for is a pair $\langle \Omega \in \mathbf{Set}, \Theta \subseteq \Omega \rangle$ such that for every other pair $\langle X \in \mathbf {Set}, Y \subseteq X \rangle$ there exists a unique morphism
$$f \in \mathbf{Set}^\text{op}(\Omega,X) = \mathbf{Set}(X,\Omega)$$
such that
$$\mathcal P (f)(\Theta)=f^{-1}(\Theta)=Y\ .$$
If we take
$$\Omega = \mathcal P(\{\emptyset\})=\{\emptyset,\{\emptyset\}\}=\{0,1\}$$
what we get is the $$\mathbf{Set}^\text{op}(\Omega,X)=\mathbf{Set}(X,\Omega)$$
is the set of characteristic functions of $X$.
For each $Y \subseteq X$ there's a unique $f \colon X \to \Omega$ such that $f^{-1}(1) = Y$, but this means exactly that for every pair $\langle X \in \mathbf{Set}, Y \in \mathcal P(X)\rangle$ there exists always a unique $f \in \mathbf{Set}^\text{op}(\Omega,X)$ such that $\mathcal P(f)(1)=Y$, and thus $\langle \Omega,1\rangle$ is a universal element for the functor $\mathcal P$.
I hope this answer may help you.
(Edit:) I want to add something else. Universal elements of a functor $\mathcal F \colon \mathbf C^\text{op} \to \mathbf{Set}$ are important also for another reason: via yoneda-lemma universal elements are those objects corresponding to natural isomorphism from functor $\mathbf C(-,\Omega)$ to the functor $\mathcal F$.
So a functor $\mathcal F$ has a universal element if and only if it is representable. This is important because representable functors actions can be represented in term of the category $\mathbf C$ itself: for instance for every morphism $f \in \mathbf C(c,c')$ we can represent the function $\mathcal F(f) \colon \mathcal F(c) \to \mathcal F(c')$ as the function
$$ g \in \mathbf C(c',\Omega) \mapsto g \circ f \in \mathbf C(c,\Omega)$$
identifying sets $\mathcal F(c)$ and $\mathcal F(c')$ with sets $\mathbf C(c, \Omega)$ and $\mathbf C(c',\Omega)$.
Best Answer
Observe that under the prescription of $\alpha_X$ in your question the following statements are equivalent:
Here every corresponds with surjectivity and unique with injectivity.
Does this make things more clear for you?