number theory - Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers?

Is there a short answer to this question, to get the overview? I once had a lecture about this topic, but it took several weeks to proof the prime number theorem using $\zeta$, so I lost track of the general idea.