The title of my Q. says it all:
QUESTION: Who defined and who coined: module?
Would it be Emmy Noether?
EDIT In view of @anon's and KConrad's answers, and as it could have been expected, the situation is a bit complex. Thus it looks that while Dedekind coined module, the final notion was defined by Emmy Noether. By defined I mean that module at a minimum should be a generalization of linear spaces, abelian groups and ideals.