I've been learning about Dedekind zeta functions and some basic Lfunctions in my introductory algebraic number theory class, and I've been wondering why some functions are called Lfunctions and others are called zeta functions. I know that the zeta function is a product of Lfunctions, so it seems like an Lfunction is somehow a component of a zeta function (at least in the case of Artin Lfunctions, they correspond to specific representations). Is this the idea behind the distinction between "zeta function" and "Lfunction"? How do things generalize to other kinds of zeta and Lfunctions?

Let me say first that a Dedekind zeta function is always a product of Artin Lfunctions. It is the structure of the Galois closure which is relevant here. Let me give a nice example which is indicative of the general case. Let $p(x) \in \mathbb{Z}[x]$ be an irreducible cubic, and let $\alpha$ be a root of $p$. Then $K=\mathbb{Q}(\alpha)$ has trivial automorphism group, and its Galois closure (say $L/\mathbb{Q}$) is an S3extension. The group S3 has three irreducible representations: the trivial representation, the "sign representation" $\chi$ which is also onedimensional, and an irreducible twodimensional representation which we will call $\rho$. Then we have the relations $\zeta_K(s)=\zeta_{\mathbb{Q}}(s)L(s,\rho)$ and $\zeta_L(s)=\zeta_{\mathbb{Q}}(s)L(s,\chi)L(s,\rho)^2$. The proofs of these facts are part of the formalism of Artin Lfunctions. Generally, the distinction is really a matter of history. Certain objects were named zeta functions  HasseWeil, Dedekind  while Dirichlet chose the letter "L" for the functions he made out of characters. However, one feature is that "zeta" functions tend to have poles, and they often "factor" into Lfunctions. These vagaries are made more precise in various places, for example IwaniecKowalski Ch. 5 and some survey articles on the "Selberg class" of Dirichlet series. 


Although noone else seems to have suggested this, my personal take on this is that there's no difference whatsoever. Sure, if people start talking about "Dedekind zeta functions" or "Artin $L$functions" then there starts to be relations amongst these specific choices. But for me, all of these are special cases of automorphic $L$functions, which, in a parallel universe, could just have easily have been called automorphic zetafunctions. History tells us that varieties have zeta functions, Dirichlet characters have $L$functions, number fields have zeta functions, elliptic curves have $L$functions and so on. But they're all just instances of the same thing really (at least conjecturally)they're all just (simple combinations of) automorphic $L$functions, which, as I say, could easily have been called automorphic zetafunctions. 


Lfunctions depend on characters (or representations), zeta functions do not (or correspond to a trivial character). For example, $L(s,\chi) = \sum_{n = 1}^{\infty}\chi(n)n^{s}$ where $\chi$ is a Dirichlet character. Supposing that $\chi_0$ is the trivial character modulo $q$, we get $L(s,\chi_0) = \zeta(s)\prod_{pq}(1  p^{s})$ where $\zeta(s)$ is the Riemann zeta function. So the Dirichlet Lfunctions generalize the Riemann zeta function. The Dedekind zeta function also generalizes in the same way, to Lfunctions with Hecke Grössencharacters. (Removed a false statement at the end; as David Hansen points out, one can get a factorization into Artin Lfunctions (belonging to the Galois closure of the extension) even when the extension is not Galois by factorizing the Dedekind zeta function of the Galois closure and taking away some factors) 


I do like the other answers, too, but it seemed silly to append comments to all... : To my perception, first, I tend to not feel a difference between "zeta function attached to ..." and "Lfunction attached to..." if only because usage is variable. Second, in many settings (analytic/automorphic or geometric/motivic or...) a zeta function is an Lfunction with relatively trivial "further data", whatever that means in context. So, Dedekind zeta functions are Hecke Lfunctions with trivial data, for example. Analogously for schemes without or with nontrivial sheaf, in the other world. This general rule is certainly not strict... depending on usage. Third, there are the systematic, partly proven, partly conjectural, miracles that "larger" zetas factor into "smaller" Lfunctions. Classfield theory and such. This does highlight the ambiguity in "usage", namely, that some "base" is necessary to understand "triviality", etc. Edit: again significantly contingent on "usage"... If we say that an "Lfunction" (or "zeta function") "has an analytic continuation (provable by us)", then this would accidentally disallow HasseWeil zeta/Lfunctions of general varieties/schemes/whatever, because in this year we know "few" cases wherein we can prove this, although conjecturally it is mostlyalways so (meaning that poles, if any, are finite and describable). Similarly, factorization into Artin Lfunctions is in one way completely fine (for decades), but, in another, unsatisfactory since we do not know their holomorphy, ... so might decide that they're not yet (in 2012) fullylegitimate "Lfunctions"? And/or that the "factorization" of Dedekind zetas into such things is not entirely satisfactory (as in a comment). I would not be surprised that such "technicalities" persist in things I know less about. :) 


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 $\frac{\zeta(s)\zeta(s1)}{L(s,E)}$, 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 $H^{2d}$. 

