After consulting with the bandersnatch department to determine exactly what the problem is, you eagerly hurry back to your office, and plunge into the task. Some weeks later, you have not been able to come up with any algorithm substantially better than searching through all possible designs. This would involve years of computation time for just one set of specifications. You certainly don't want to return to your boss's office and report:
I can't find an efficient algorithm, I guess I'm just too dumb.
To avoid serious damage to your position within the company, it would be much better if you could prove that the bandersnatch problem is inherently intractable, that no algorithm could possibly solve it quickly. Then you could stride confidently into the boss's office and proclaim:
I can't find an efficient algorithm, because no such algorithm is possible.
Unfortunately, proving inherent intractibility can be just as hard as finding efficient algorithms.
The theory of NP-completeness provides many straightforward techniques for proving that a given problem is "just as hard" as a large number of other problems that are widely recognize as being difficult and that have been confounding the experts for years. Armed with these techniques, you might be able to prove that the bandersnatch problem is NP-complete and march into your boss's office and announce:
I can't find an efficient algorithm, but neither can all these famous people.
At the very least, this would inform your boss that it would do no good to fire you and hire another expert on algorithms.
Now you can spend you time looking for efficient algorithms that solve various special cases of the general problem. You might look for algorithms that, though not guaranteed to run quickly, seem likely to do so most of the time. Or you might even relax the problem somewhat, looking for a fast algorithm that merely finds designs that meet most of the component specifications. Thus, the primary application of the theory of NP-completeness is to assist algorithm designers in directing their problem-solving efforts toward those approaches that have the greatest likelihood of leading to useful algorithms.