Zeta functions arise from schemes. $L$-functions arise from schemes + a sheaf on that scheme.
Obviously, the zeta function of a number field $K$ is the zeta function of $\operatorname{Spec} K$.
The $L$-function of an elliptic curve is the $L$-function of its Tate module. Its zeta function is $\frac{\zeta(s)\zeta(s-1)}{L(s,E)}$, where $\zeta(s)$ is the Riemann zeta function.
The factorization of a zeta function into $L$-functions is the factorization of the etale cohomology into irreducible Galois representations. The poles arise from the factors that are Tate twists of trivial representations, in particular from $H^0$ and $H^{2d}$.
