The responsibility of a proofreader lies in the name: proof reader. A proof is a completed document that is ready for publication. The proofreader compares the finished document with the edited manuscript to ensure all errors have been fixed and that all formatting is correct. Before the document reaches the proofreader, however, it must go through the editing process.
There are seven roles in the book editing process - book coach, beta reader, manuscript evaluator, developmental editor, line editor, copyeditor, and proofreader - but these roles can be condensed or disregarded for other kinds of texts. The writer of an academic paper, for example, would not utilize the services of a book coach or a beta reader, but they might ask a peer to review it against their instructor's standards.
In the simplest terms, book coach, beta reader, manuscript evaluator, and developmental editor are all content editors. They look at the substance of the story or article. The line editor, who can also be the developmental editor, reads for literary style and tone. The copyeditor looks at the mechanics of the text, fixing spelling, grammar, punctuation, capitalization, hyphenation and compound words, numbers, trademarks, copyright and permissions, word choice, and some text formatting. They also do basic fact-checking and, for books or serial projects, create style sheets.
After a manuscript - book or paper - has been through the editors and all the corrections have been made, it is ready for publication. The proofreader gives the finished document one last pass.