One basic problem in mathematics is to solve equations. Giving a polynomial non-trivial of complex coefficients on one variable, we know that it has as many zeros as its degree. The simplest holomorphic function which has infinite zeros is perhaps the sinus and cosinus functions.
Of course, from complex analysis we know that a holomorphic function on the whole plane can not have infinite zeros on a bounded region. So we can talk about the number of zeros of a holomorphic function inside a circle of certain radius.