Motivation for the Definition of Compact Space A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} \subseteq \mathcal{A}$ such that $C \subseteq \bigcup_{U'\in \mathcal{A'}} U'$.
Now, this definition leads to many interesting results, but if I were teaching someone about compact sets, how would I motivate this? Concepts like sequential compactness, open and closedness, and even connectedness are reasonably easy to motivate. I can not see how to motivate this definition. Compact spaces are often seen as generalizations of finite spaces. They are also seen as a generalization of boundedness and closedness. I can't see how to connect the definition with these concepts.
Alternatively, is there a definition of a compact set which is easier to motivate?
 A: According to Munkres, the original definition of compactness is a space which satisfies the Bolzano-Weierstrauss property holds.  That is, if every infinite subset has a limit-point.  
Unfortunately, it turns out, this conception of compactness,sometimes called limit point compactness, doesn't have all the useful properties that compactness has.
For example, the continuous image of a limit point compact space need not be limit point compact.  Also, a limit point compact subspace of a Hausdorff space need not be closed.
A: An equivalent def'n is that if $ F$  is a non-empty family of closed sets with the F.I.P. (Finite Intersection Property) then   $\cap F \not = \phi $ .  This generalizes the idea of limits , and you can show that many results, e.g. on bounded closed subsets of $ R^n$ , using this property, so it is seen to be a useful tool that a space is compact. Once you show some additional consequences, e.g. that a continuous image of a compact set is compact, you can show how to apply them, e.g. in analysis, showing that an extremum exists, (hence the Mean Value Theorem in calculus).  So you get easier results and new ones, from the compactness. 
A: One of my favorite textbooks is Klaus Janich's Topology, and he has a nice motivation for compactness I feel, namely why we should care about. This is in addition to my comment about compact subsets of a Hausdorff space being essentially like finite point sets. But he writes:

In compact spaces, the following generalization from "local" to "global" properties is possible: Let $X$ be a compact space and $P$ a property that open subsets of $X$ may or may not have, and such that if $U$ and $V$ have it, then so does $U\cup V$. Then if $X$ has this property locally, i.e. every point has a neighborhood with property $P$, then $X$ itself has property $P$.

This is nice, but it is slightly advanced, and he gives some examples that follow like a continuous/locally bounded map from a compact space to $\mathbb{R}$ is bounded, and some discussions of locally finite covers and manifolds (honestly, I like this book after the fact of learning topology, not to learn from).
Hope that helps somewhat.
A: For boundedness:
You can give an exercise so that the students need to show that a bounded metric can induce the same topology as an unbounded one (at least you can easily show that for metric spaces with $d(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$). So, boundedness is not really a topological property.
If each open covering has a finite subcovering (and using that bounded metric instead of the regular one), you can associate compact sets with sets that are not too big, having something like an idea of boundness. In fact, they behave "pointwise" ($T_2$ and compact implies $T_4$, $f(K)$ compact again, for $K$ being compact, etc.)
As you are asking for a motivation, I think the metric space should be fine.
A: Topological spaces are made out of open sets.
Sometimes, you have the occasion to write your topological space as a union of open sets: e.g. because whatever you are trying to study is easy to understand when restricted to just one of the sets.
If you can motivate that it is useful to do this sort of thing, then the usefulness of the usual definition of compactness is almost self-evident; a finite union is much easier to work with than an infinite union.
A: The motivation for the definition of compactness is that that condition is extremely useful. Essentially every proof of every fact about the Riemann integral on the line, for example, depends on it.
Definitions capture useful properties which allow us to prove useful things — the good ones, at least.
If you want to motivate the definition, give the definition and immediate proceed to prove useful things with it. If you are looking for an a priori reason that justifies the definition, well, there is none.
A: I think the better way to motivate compactness is in real analysis, a set $F$ is said be a compact set if it bounded and closed. Is very easy to imagine something compact like this one. The general definition, in general topological spaces, is motivate by Bolzano-Weierstrass theorem.
A: Well I like to think of compactness like this because it gives me the idea of a compact set in terms of closed sets. 
A collection $\mathcal{C}$ of subsets of $X$ is said to have the finite intersection property if for every finite subcollection 
$$\{C_1,\cdots C_n\}$$ 
of $\mathcal{C}$ , $\bigcap_{i=1}^{n} C_i$ should be nonempty.
Now if in a topological space every collection $\mathcal{C}$ of closed sets in $X$ have the finite intersection property , then the space $X$ is compact.
Well instinctively you would always think of a compact set as a closed set(though we have examples where compact sets are not closed.. One lies in your backyard!!)  here is a definition involving closed
 sets.
Also you can use this definition to prove the tychonoff theorem 
A: A compact set is the "next best thing to finite"- every compact set is both closed and bounded (in a metric space), just like a finite set.   In a metric space, given any point, p, there are both minimum and maximum distances from p to points in the compact set.
