Why is it important to prove that a problem is NP-complete? Am I correct in understanding that proving a problem NP complete is a research success? If so why?
Ali, good question.  

Suppose you want to show that some problem P is computationally hard.  Now, you could conjecture that P is hard just based on the fact that we don't have any efficient algorithms for it yet.  But this is rather flimsy evidence, no?  It could be that we have missed some nice way to look at P which would make it very easy to solve.  So, in order to conjecture that P is hard, we would want to accumulate more evidence.  Reductions provide a tool to do exactly that!  If we can reduce some other natural problem Q to P, then we have shown P is at least as hard as Q.  But Q could be a problem from some completely different area of mathematics, and people may have struggled for decades to solve Q also.  Thus, we can view our failure to find an efficient algorithm for Q to be evidence that P is hard.  If we have lots of such Q's from many different problem domains, then we have a huge body of evidence that P is hard.

This is exactly what the theory of NP-completeness provides.  If you prove your problem to be NP-complete, then you have tied its hardness to the hardness of hundreds of other problems, each of significant interest to various communities.  Thus, morally speaking, you can be assured that your problem is indeed hard.