How to understand compactness? How to understand the compactness in topology space in intuitive way? 

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.)
