We talk about three rather different but not unrelated conjectures here:
(1) Artin $L$-functions are automorphic $L$-functions;
(2) the Selberg class consists of automorphic $L$-functions;
(3) automorphic $L$-functions belong to the Selberg class.
The three families of $L$-functions occurring here are defined very differently. Artin $L$-functions are defined in terms of Galois representations, automorphic $L$-functions are defined in terms of automorphic representations, while the Selberg class is defined via natural axioms of an analytic nature. Conjectures (1) and (3) are instances of the Langlands conjectures, while (2) strengthens the idea that sufficiently nice analytic properties of a Dirichlet series are always "caused by" an automorphic form (or automorphic representation) behind the Dirichlet series.