To every local admissible representation $\pi_v$ of $\mathrm{GL}_n(k_v)$, there is a local $L$-function $L(s,\pi_v)$. For a global admissible representation $\pi=\otimes_v \pi_v$ of $\mathrm{GL}_n(\mathbb{A}_k)$, the corresponding global $L$-function is defined as $L(s,\pi)=\prod_v L(s,\pi_v)$. However, unless $\pi$ is automorphic, this global $L$-function will not have the usual analytic properties (cf. converse theorems). So it is very special for a global admissible representation to be automorphic. Automorphicity of $\pi=\otimes_v \pi_v$ connects the various local factors $\pi_v$. In particular, if you change finitely many local factors $\pi_v$ in a given cuspidal automorphic representation $\pi=\otimes_v \pi_v$, the resulting global admissible representation will no longer be cuspidal automorphic (cf. multiplicity one theorems).
BTW I recommend Goldfeld-Hundley's two-volume textbook on automorphic forms (in the same series as Bump's textbook). Also, there is this related earlier post of mine that you might find useful.