How to understand compactness? How to understand the compactness in topology space in intuitive way? 
 A: The way to understand compactness is to see it in action. As you learn more, you'll see more and more situations in which compactness is useful, even fundamental. With the accumulation of evidence, like geological layers, you will construct understanding. Then one day you'll come across a new way of using compactness, a new angle, and then you will see that in fact you only had understood part of it... and this should keep going.
I am a firm believer that asking for understanding and, much worse, for intuitive understanding of things when one more or less has just encountered them is not the correct way to reach understanding. 
A: I too was looking for a good motivation of compactness till I came across this article written by Terry Tao which was suggested by Phil Ellison on MO. The article is a good read and is really illuminating. The article can be found here.
A: For nice enough spaces, e.g. metric spaces, compactness is equivalent to sequential compactness. I like to think of sequential compactness as saying that sequences of points can't run off to "infinity".
A: Maybe you should think about compactness, as something that takes local properties to global properties. For example, if $f:K\rightarrow \mathbb{R}$ is continuous, $K$ is compact, and $f(x)>t_x>0$ for all x, then you can find $t>0$ such that $f(x)>t>0$ for all x - so from $f(x)>t_x>0$ point wise, you know that $f>t>0$ as a function. (This is a simple consequence of Weierstrass theorem in $[a,b]\rightarrow \mathbb{R}$)
Usually we find some property that is true for every "small" enough open sets, then use compactness to reduce the case to finitely many open sets and use induction to show that the property is true for all of the space. This is at least how I understand compactness.

As the commenters below this message wrote (and I didn't emphasize enough), we usually use compactness to reduce infinite problems\conditions\restraints to a finite subset that cover the entire space, and then use some argument that works only for finite cases (like induction, taking max\min, take finite sums etc). In my example above, we wanted to find a minimum over all the lower bounds, but in the infinite case this is usually just an infimum (and can be zero), but when we reduce to a finite case there is a minimum.
In this way we can think of compactness as something that let us use some finite argument on infinite covers (and many times to transfer some property from the cover to the entire space). 
A: As I mentioned in another question, I tend to intuit compactness as a sort of "super closedness."  There are a few reasons for this.
First, in Hausdorff spaces, every compact subset will automatically be closed.
Second, in first-countable spaces, one can think about "closedness" as "closed under limit operations."  That is, a set $E$ is closed if and only if it contains the limits of all sequences in $E$.  By analogy, every compact subset of a first-countable space is sequentially compact -- that is, every sequence has a convergent subsequence.  (Note, however, that the converse of this is not generally true in first-countable spaces.)
Third, in metric spaces, compactness is equivalent to being complete and totally bounded.  This can be viewed as a generalization of the Heine-Borel Theorem which classifies the compact sets of $\mathbb{R}^n$.  In such a scheme, completeness corresponds to closedness, and total boundedness to boundedness.
Note that metric spaces are automatically Hausdorff and first-countable, so all of the above certainly applies to metric spaces.

For general topological spaces (without any assumption of "Hausdorffness" or first countability), one can also intuit compact topological spaces as those with relatively few open sets.  Indeed, if a set $X$ is compact under a given topology, then it will also be compact under any weaker (coarser) topology.

EDIT: To add to Prometheus' answer, I would also say that compactness is a property that plays especially well with continuity (Extreme-Value Theorem, Heine-Cantor Theorem) and convergence of functions (Dini's Theorem).  As Prometheus said, this is in part because of its role in reducing global problems to local ones -- or, perhaps, infinite ones to finite ones.
(I'm thinking particularly of the proof of Heine-Cantor which relies on the fact that the infimum of a finite number of positive distances is positive -- whereas an infinite number of positive distances may have an infimum of zero.)
A: Look at as many examples of what sets are compact and which ones are not, then you get a feeling of what compactness tries to encapsulate. Then try to see why the examples of compact sets you looked at are compact and why the non compact examples are not. Human mind learns a lot by comparing and mixing stuff, so just compare mix, add repeat.  
A: I think of compactness as topological finiteness.  A real valued function defined on a finite set has a maximum and a minimum.  A continuous real valued function defined on a compactum has a maximum and a minimum.  It is the finite subcover property that is at the heart of this intuition.
A: For me, the compactness of a topological space means that it has enough points to provide exact solutions to continuous equations. More precisely,

compactness = Any equation that can be approximated by a consistent system of $\leq$ inequalities of continuous functions has a solution.

For instance, being a solution to the equation $x^2 = 2$ is equivalent to being a simultaneous solution to the infinite system of inequalities $\lbrace |x^2-2| \leq \frac 1 n \rbrace_{n\in\mathbb N}$. This system is consistent in the sense that every finite subsystem has solutions; after all, we can approximate $\sqrt 2$ by rational numbers arbitrarily well. Then, it is the compactness of bounded intervals of real numbers that tells us that there exists a simultaneous solution of all of these inequalities, and hence of the equation itself.
In contrast, trying to encircle $\sqrt 2$ as a rational number fails, you can enclose it by smaller and smaller intervals, but when you try to catch it with an infinite number of intervals, it "falls through the cracks".
The principle discussed above is also known as the finite intersection property. The inequalities, or "conditions", define a collection of closed sets and the exact solution we seek is contained in the intersection of infinitely many closed sets. If every finite intersection of these closed sets is non-empty, i.e. if every finite collection of conditions can be fulfilled, then one could expect that all of the conditions can be satisfied simultaneously. This is precisely what compactness ensures.
Other examples:


*

*Peano's theorem about the existence of solutions to ordinary differential equations relies on compactness to construct a good candidate.

*A commonly used technique for showing the existence of solution for partial differential equations is the calculus variations. In this context, the weak compactness of the unit ball in a Hilbert space is often used to get a weak solution to the PDE which is then upgraded to a strong solution.

