Developmental editors work with the author to craft the manuscript, looking at structure and argument in non-fiction or plot and character in fiction. (In traditional publishing, these are usually the acquiring editors.)
Line or substantive editors also look at the manuscript as a whole, but generally don’t work as closely with the author and aren’t expected to edit as deeply. (This and the previous category are sometimes lumped together as substantive editing.)
Copy editors concentrate on the language or copy. They focus on trying to make the style of the manuscript clean and consistent.
Proofreaders are usually the last folks who look at a book, in galley or proof form, as it’s about to go off to be printed (or, in the case of ebooks, as it’s about to enter distribution). They’re looking purely for misspellings or errors in style, such as improper punctuation, grammar or formatting.