Why is the notion of analytic function so important? I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of "analytic function" so important? 
I guess being analytic entails some more interesting knowledge rather than just that it can be approximated by Taylor power series, right? 
Or, maybe I don't understand (underestimate) how a Taylor power series is important? Is it more than just a means of approximation?

As Chappers says, the analytic property of a function is very useful on those defined on the complex plane, and it turns out that all the usual functions are analytic.
Those functions have very interesting properties, such as complex-derivative, zero integral on closed paths, and the residue formula. You can also use a lot of real analysis results (Leibniz' formula, the chain rule) in the study of analytic functions.
For instance, without complex-analytic related tools, it would be impossible to prove major theorems like the Prime Number Theorem. As a matter of fact, it is quite astounishing that properties of complex function need to be used to prove a result about arithmetics.
