What do I need to know to understand the Riemann hypothesis

Which kinds of fields of mathematics do I have to know about in order to understand the Riemann hypothesis millenium prize problem?

Just to understand the$^\dagger$ statement of the problem, you would have to be familiar with complex analysis and analytic number theory. The $\zeta$ function itself is an analytic object from number theory and to understand its significance (just on the surface!) you would have to study it in these realms. Of course it is also a function on $\Bbb C$ after analytic continuation - attained using a functional equation - with a simple pole at $1$, and understanding what this means and how to manipulate the function deftly will mean studying complex analysis.

$^\dagger$I refer to the statement that $\zeta(s)$ has all nontrivial zeros on the critical line. There are actually a lot of equivalent statements that require very little knowledge of complex analysis (you'll still need to pick up a few definitions of arithmetic functions from analytic NT for many of them, these aren't too hard). You can find a lot of equivalences listed here for example.

Beyond that, to understand the modern approaches to RH and related or generalized conjectures and all of the theory there is surrounding this creature, you must go much further in algebraic number theory at the very least, and travel to many other worlds like modular forms, differential geometry, quantum theory and random matrices, etc. - basically at least a basic knowledge of most advanced subjects in analysis, algebra and geometry, and then especially deeply in pertinent areas.

I used to try to explain the complex analysis part. I have given up on that. See http://en.wikipedia.org/wiki/Prime-counting_function for details...

The short version is that the prime counting function $\pi(x)$ is approximated pretty well by $$ \frac{x}{\log x}. $$ It is known that a better approximation is given by $$ \mbox{li} \, (x). $$

The Riemann hypothesis is equivalent to a specific form of the statement that $ \mbox{li} \, (x) $ is a really good approximation, see http://en.wikipedia.org/wiki/Prime-counting_function#The_Riemann_hypothesis and TABLE

Definitely, the number theory. You can find more connections to other fields in "The Millenium Problems" by Keith Devlin.

The Riemann Hypothesis is just a conjecture about the zeros of a function. Understanding this is simple.

However, understanding why this is important and what it means to number theory is far from simple.