First assume every submodule of M, including M itself, is finitely generated. Let H1 H2 H3 … be an infinite ascending chain of submodules, with H the union over Hi. Now H is a submodule of M, and is spanned by a finite set of generators. All these generators live in some submodule Hn, and beyond that, all submodules in the chain are equal to Hn. The infinite ascending chain cannot exist, and M is noetherian.
Conversely, assume H cannot be spanned by a finite set of generators. Select x1 in H and let H1 be the submodule generated by x1. This is not all of H, so select any x2 in H-H1 and let H2 be the submodule generated by x1 and x2. Once again these two generators cannot span all of H, so select any x3 in H-H2 and let x1 x2 and x3 span H3. This process continues forever, building an infinite ascending chain. Therefore M is noetherian iff every submodule is finitely generated.
Every ring with finitely generated ideals, such as a pid, is noetherian.