Complex Analysis applications toward Number Theory

This question is like asking how abstract algebra is useful in number theory: lots of it is used in certain areas of the subject so there's no tidy answer. You probably won't be using Morera's theorem directly in number theory, but most of single-variable complex analysis is needed if you want to understand basic ideas in analytic number theory. A few topics you should pay attention to are: the residue theorem, the argument principle, the maximum modulus principle, infinite product factorizations (esp. the Hadamard factorization theorem), the Fourier transform and Fourier inversion, the Gamma function (know its poles and their residues), and elliptic functions. Basically pay attention to the whole course! There really isn't a whole lot in a first course on complex variables where one can say "that you should ignore if you are interested in number theory".

If you want to be careful and not just wave your hands, you need to know conditions that guarantee the convergence of series and products of analytic functions (and that the limit is analytic), the existence of a logarithm of an analytic function (it's not the composite of the three letters "log" and your function), that let you reorder terms in series and products, that justify termwise integration, and of course the workhorse of analysis: how to make good estimates.

I think basic is on the right track. The two big classical theorems in analytic number theory whose classical proofs use some complex analysis are Dirichlet's Theorem on primes in arithmetic progressions and the Prime Number Theorem. (It is also useful to learn about the combination of the two: the Prime Number Theorem for Arithmetic Progressions.)

For the former, I can recommend my own lecture notes:

http://math.uga.edu/~pete/4400dirichlet.pdf

http://math.uga.edu/~pete/4400DT.pdf

The second part is explicitly a digested version of the proof Serre presents in his Course in Arithmetic. I don't have a similarly canonical reference to give you for the proof of the Prime Number Theorem (i.e., I don't have any notes on it!), but it can be found in many analytic number theory books, for instance in Apostol's Introduction to Analytic Number Theory, Davenport's Multiplicative Number Theory or G.J.O. Jameson's The Prime Number Theorem.

I'll second Pete Clark's answer, and note that there are some other big theorems in analytic number theory for which there are proofs using Complex analysis. For example, there is the asymptotic formula for the number of partitions of $n$, which formula is ${e^{\pi\sqrt{2n/3}}\over4\sqrt3n}$. There is a proof in Donald J Newman, Analytic Number Theory, but be warned that the chapter on the partition function is infested with typos.