Why study Proper Maps? We invite the reader to recall that a proper map $f: X \to Y$ between topological spaces $X,Y$ is a continuous map such that for all compact sets $K \subset Y$, $f^{-1}(K) \subset X$ is compact. 
Proper holomorphic maps play a large role in the theory of holomorphic functions of several complex variables, of which I am beginning my studies. 
I am curious as to why we study these maps in particular? I am aware of the proper mapping theorem, which asserts that the image of an analytic space under a proper holomorphic mapping is an analytic subvariety of the target analytic space, but I'm not really seeing the significance of these results. 
Any motivation or insight is appreciated. Thanks. 
 A: This answer only addresses the (general-topology) aspect of your question.
The topological theory of proper mappings is very rich and full of interesting invariants, the simplest of which is the set of ends. Given a connected manifold $X$ (or more generally a path connected locally compact space), the collection of subsets $U \subset X$ such that $X-U$ is compact forms an inverse system under inclusion, and the inverse limit of this set is, by definition, the set of ends $\text{Ends}(X)$. This defines part of a functor from the category of connected manifolds and proper maps to the category of sets and functions: for any proper map $f : X \to Y$ there is an induced function $f_* : \text{Ends}(X) \to \text{Ends}(Y)$. 
As an application, if $Y$ has more ends than $X$ in the sense of cardinality then there are no surjective proper maps from $X$ to $Y$, because the map on ends induced by a surjective proper map is also surjective, as one can check.
One can generalize ends to a richer collection of invariants in the category of proper maps, namely the cohomology with compact supports $H^n_c(X;\mathbb{R})$. This is connected with the number of ends by the theorem that the rank of $H^1_c(X;\mathbb{R})$ is equal to one plus the number of ends (I think I have that right).
A: This is an intuitive idea on proper maps between metric spaces; and it is extendible to topological spaces which admits an Alexandroff compactification satisfying the Hausdorff's separation axiom.

Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces, let $f:X\to Y$; $f$ is continuous if and only if for any convergent sequence $\{x_n\in X\}_{n\in\mathbb{N}}$, $\{f(x_n)\in Y\}_{n\in\mathbb{N}}$ is convergent.
Otherwise, let $\{x_n\in X\}_{n\in\mathbb{N}}$ be a sequence escape to infinity, that is
$$
\forall n\in\mathbb{N},\,\exists K_n\Subset X\mid\{x_1,...,x_n\}\subseteq K_n,\,\forall m>n,\,x_m\notin K_n.
$$
It is easy to prove that $f$ is proper (as the OP's definition) if and only if for any sequence $\{x_n\in X\}_{n\in\mathbb{N}}$ escapes to infinity, also $\{f(x_n)\in Y\}_{n\in\mathbb{N}}$ escapes to infinity.

A little remark: In algebraic geometry, there exists a notion of proper morphism, which is more complicated as one can read here.
A: Proper maps preserve holomorphic convexity. That is, if $D$ is holomorphically convex and $f$ is proper, then $f(D)$ is holomorphically convex. 
