NP-completeness 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), NP-completeness of a problem means that no polynomial-time algorithm is known, but that statement may eventually become false ^{*}. Maybe it's better to say “Computer scientists believe that, if a problem is NP-complete, then there is no polynomial-time algorithm to solve it”?
Also, I'm not sure that it's fair to say that NP-completeness of a problem means that the time-complexity of the problem grows exponentially in the input. Again, we think that NP-completeness correlates with exponential time-complexity, but that could change ^{*}. For that matter, can't NP-complete problems have super-exponential complexity (like 2^(n^2))—or are you using ‘exponential’ in the generic sense of ‘faster-growing than polynomial’?
^{*} Although we all know that it won't really. :-)