In page 27 of Katok's Fuchsian Groups, the following definitions are given.
A family $\{ M_{\alpha} \: \vert \: \alpha \in A \}$ of subsets of $X$ indexed by elements of a set $A$ is called locally finite if for any compact subset $K \subset X, M_{\alpha} \cap K \neq \emptyset$ for only finitely many $\alpha \in A$.
We say that a group $G$ acts properly discontinuously on $X$ if the $G$-orbit of any point $x \in X$ is locally finite.
I feel I am stupid for asking this but we have defined the concept of locally finite for set of subsets of $X$. But $G$-orbit of a point $x \in X$ is a subset of $X$. Then what does "$G$-orbit of any point $x \in X$ is locally finite" in definition of properly discontinuous action mean?
Local finiteness here simply means that for every $x\in X$ and every compact $K\subset X$ the set $$ \{g\in G: gx\in K\} $$ is finite. (The $G$-orbit $Gx$ is not just an element of the powerset of $X$, but is an indexed subset of $X$ with the index set $G$.)
This said, Katok's definition is faulty, as it was discussed at MSE many times. In particular, it does not guarantee Hausdorffness of $X/G$. Katok's definition is intended for use only in the case of isometric group actions.
The true definition of proper discontinuity for topological group actions is different, it requires finiteness of $$ \{g\in G: gK\cap K\ne \emptyset\} $$ for every compact $K\subset X$.