What is so interesting about the zeroes of the Riemann $\zeta$ function? The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$

The Riemann hypothesis asserts that all the non-trivial zeroes of the $\zeta$ function  lie on the line $\text{Re}(s) = \frac{1}{2}$. 

My question is:

Why are we interested in the zeroes of the $\zeta$ function? Does it give any information about something?

What is the use of writing $$\zeta(s) = \prod_{p} \biggl(1-\frac{1}{p^s}\biggr)^{-1}$$

Here is a visual supplement to Eric's answer, based on this paper by Riesel and Göhl, and Mathematica code by Stan Wagon:

The animation demonstrates the eventual transformation from Riemann's famed approximation to the prime counting function
$$R(x)=\sum_{k=1}^\infty \frac{\mu(k)}{k} \mathrm{li}(\sqrt[k]{x})=1+\sum_{k=1}^\infty \frac{(\log\,x)^k}{k\,k!\zeta(k+1)}$$
to the actual prime-counting function $\pi(x)$, through a series of successive corrections based on the nontrivial roots of $\zeta(s)$. (Here, $\mu(k)$ is the Möbius function and $\mathrm{li}(x)$ is the logarithmic integral.) See the Riesel/Göhl paper for more details.
