First of all, one shouldn't distinguish too much between the terms $L$-series and $\zeta$-function; it is more or less a matter of history which term you use in a given context.
The true distinction is between objects on the "motivic" (Diophantine equation or algebraic number theory) side, and objects on the "automorphic" side.
So, Artin $L$-functions, and $\zeta$-functions and $L$-functions attached to varieties, are on the motivic side. They don't have obvious analytic continuations or functional equations.
Dirichlet $L$-functions (i.e. the ones built out of characters of the multiplicative group mod $N$ for some $N$), Hecke $L$-functions (built out
of ideal class group characters), $L$-functions of modular forms, or more generally automorphic forms, are automorphic $L$-functions. Except for the
last (general automorphic $L$-functions) these can be proved to have analytic continuation with functional equations. The natural limit of these analytic continuation/functional equation arguments (which begin with Riemann, with Tate's thesis being another major point en route) is the fact that standard $L$-functions for automorphic forms on $GL_n$ have analytic continuation/functional equations.
Langlands's functoriality conjecture in particular conjectures that any (a priori more general) automorphic $L$-function is in fact a standard $L$-function. Many special cases are known (and Ngo got the Fields medal for proving the fundamental lemma, which is one tool in proving certain cases, namely those arising from endoscopy), but it is wide open in general.
The reciprocity conjecture then states that any of the motivic $L$-functions are also actually standard $L$-functions. (E.g. for $L$-functions of elliptic curves over $\mathbb Q$, this becomes the statement that any such elliptic curve is associated to a weight 2 modular form, which was proved by Wiles et. al.) This is also wide open in general.
The special case of Artin $L$-functions, and some other special cases, are actually covered by both conjectures. (This is because, if you broaden your perspective sufficiently, as Langlands did, by allowing arbitrary reductive algebraic groups into the picture, then Artin $L$-functions can be thought of as
a particular kind of automorphic $L$-functions. This is not so easy to see when you are just entering the subject, but one can think of just the Riemann $\zeta$-function: this is certainly motivic, being the $\zeta$-function of the
variety Spec $\mathbb Q$, but is also automorphic, being the Dirichlet $L$-function for the trivial character.)
But in general, the conjectures have a slightly different nature: functoriality can be thought of (if you want) purely in terms of harmonic analysis on adelic groups, whereas by its very nature reciprocity involves arithmetic geometry. In practice, at least at the moment, the two seem to be fairly intertwined. Functoriality certainly is a tool that can be very helpful in proving reciprocity; also, of those cases of the Artin conjecture which are known, some are proved using the functoriality view-point (e.g. Langlands--Tunnell) and some using the reciprocity view-point (e.g. Buzzard--Dickinson--Shepherd-Barron--Taylor).
