Execution time of NP and NP-Complete algorithms
Asymptotic (/Big-Oh/Landau) notation is not special to polynomial time algorithms, or indeed any class of algorithm. It is the standard way of communicating (asymptotic) running times, amongst other resources.

So we say that binary search runs in time $O(\log n)$ because that is what its asymptotic running time is, and hence the Searching problem in $\mathrm{P}$, but not it is in $\mathrm{P}$ and hence we can get some $O(\cdot)$ expression for it. The Tower of Hanoi problem has an $O(2^{n})$ algorithm for example (in fact $\Theta(2^{n})$), this happens to put it in the class $\mathrm{EXP}$ , but the asymptotic expression is still an evaluation of the running time, regardless of which class it is in.

Note the distinction here between problems an algorithms as well. A problem is in a complexity class, and algorithm has a running time. By extension we say that the problem can be solved in time "blah" because it has an algorithm runnning in that time. So we can't have an $\mathrm{NP}$-complete algorithm, only an $\mathrm{NP}$-complete problem. We can abuse the language a small amount an say we have and $\mathrm{NP}$ algorithm, but what we mean is that we have  a nondeterministic polynomial time algorithm, i.e. an algorithm that demonstrates that whatever problem it's for is in $\mathrm{NP}$, just as binary search is a polynomial time algorithm that demonstrates that Searching (the formally defined problem) is in $\mathrm{P}$. Then David Richerby's answer applies - they're just polynomial but on a nondeterministic machine, i.e. $O(n^{c})$ for some fixed $c$.

On the flip side, we can have an algorithm for an $\mathrm{NP}$ problem that is not the one that shows that the problem is in $\mathrm{NP}$. For example, there's an algorithm for the Travelling Salesman Problem that just tries all the possibilities. This algorithm has running time $O(n!)$. It's not the best algorithm, but it is an algorithm for an $\mathrm{NP}$-complete problem, and that's its running time expressed asymptotically.

Finally, $\mathrm{P}$ is a subclass of $\mathrm{NP}$, so all those "P algorithms" are also "NP algorithms" - they just don't use the nondeterminism.