NPcompleteness is a property of an algorithm. It implies that no algorithm is known to solve the problem in polynomial time.
This means that if you increase the length of the input for the problem, the execution time will increase exponentially. (Of course there are input cases which are polynomial, but many of interest are not). Essentially, it means that brute force is the only known method to tackle the problem exactly.
I think that this may be a little misleading. Right now (as 6 years ago), NPcompleteness of a problem means that no polynomialtime algorithm is known, but that statement may eventually become false ^{*}. Maybe it's better to say “Computer scientists believe that, if a problem is NPcomplete, then there is no polynomialtime algorithm to solve it”?
Also, I'm not sure that it's fair to say that NPcompleteness of a problem means that the timecomplexity of the problem grows exponentially in the input. Again, we think that NPcompleteness correlates with exponential timecomplexity, but that could change ^{*}. For that matter, can't NPcomplete problems have superexponential complexity (like 2^(n^2))—or are you using ‘exponential’ in the generic sense of ‘fastergrowing than polynomial’?
^{*} Although we all know that it won't really. :)
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