Types of Editing
It may surprise you to learn that "editing" is not a single activity. I'm certified in four different types of editing: structural editing, stylistic editing, copy editing, and proofreading. As we work together to define your needs, it helps if we start from a shared understanding of the different types of work I might do on your project. Here's how Editors Canada defines each one, which is how I see it, too (you can also read it in their own words, along with several other definitions, at editors.ca).
structural editing and manuscript evaluation
Structural editing looks at your writing in broad terms. A structural editor helps you make sure your overall argument is sound, that you've organized your content in the best way possible, that you haven't left out any important points. In fiction, a structural edit evaluates the elements of your story overall—the plot, the characters, the pace, the tone, the point(s) of view, the setting, etc.
What this looks like as an editing service can vary. You can hire an editor to restructure your piece themselves, or you can have them write up a detailed report/critique, leaving you to make the changes. I call that report a "manuscript evaluation," and I recommend that all book authors in particular have one done before they launch into stylistic or copy editing. In fiction and non-fiction alike, it's important to get the foundation of the book right before you start editing it line-by-line.
In business writing projects, structural editing is often done alongside stylistic and copy editing.
stylistic editing (also called Line editing)
Stylistic or line editing is about improving the writing itself at a line-by-line level. (It is quite often combined with copy editing, although I find it's useful to define them as two separate activities as we talk through your project, because you may want one and not the other.) A stylistic editor helps to clarify your meaning, smooth out the language, make it flow better. They fix awkwardness, tighten things up, find better ways of expressing things. Often they work to eliminate jargon and adjust the reading level to make your writing appropriate to your audience.
You might say that stylistic editing can be a little subjective. Unlike copy editing, which is about following rules, or creating rules and then applying them, stylistic editing can leave much more to interpretation. Now, that's not always true. A stylistic editing step may very well involve applying rules (say, those set out by your publisher's style guide or your company's brand guidelines), such as reworking sentences to eliminate the passive voice or not using industry jargon.
But indeed stylistic editing does involve a certain amount of subjectivity, and that's why it's important that you like your editor's style. A good editor will preserve your voice as an author; you don't want someone who will completely rewrite your piece in their own voice. On the other hand, you may have distracting writing quirks you're not aware of (most of us do), and a good editor will help clear them away. Of course you'll want to make sure you like the way in which your editor handles such changes; especially in stylistic editing, it's important to get a sample edit before you begin.
The (over)simplified explanation of copy editing is that it involves fixing issues of spelling, grammar, punctuation, and other writing mechanics—in fact, this is what a lot of people think is the entirety of an editor's job. But a copy editor does much more than that: what they're primarily concerned with is ensuring consistency throughout your document. So, yes, they'll fix your mistakes, because you want your language use to be consistently correct. But also, in cases where there's no black-and-white rule, they'll help you make choices and stick to them.
Should black-and-white rule be hyphenated like that? Doesn't really matter, as long as you do it the same way every time. Your copy editor will notice if your character Theresa becomes Teresa (or Judy!) on page 47. They'll catch the spot where you cite survey responses as 56% Yes and 42% No—asking "what happened to the missing two percent?" And yes, they'll flag that you used % in one place and percent in the next.
It goes deeper. A copy editor is also concerned with the consistency of your story or argument itself: does a character pull on a pair of jeans in the morning but then later spill lunch on her skirt? Does she leave work at 5 p.m. in July in Chicago to find it's dark outside, or drive for an hour and only travel a few blocks? Do the detectives log the same piece of evidence two separate times or fail to comment on a clue that's obvious to the reader? Do you say your product is available in eight colours but only list six? A good copy editor will catch all of these issues and either fix them or flag them for you to address.
[Depending on the situation, a copy editor can also do things like indicate heading levels or placement of graphics for the designer, and in the newspaper world copy editors also write story headlines and image captions (or, they did before newsrooms started cutting copy editing...).]
A lot of people confuse copy editing with proofreading. Proofreading is actually the very last step in the editorial process, which happens not just after the manuscript is finished, but after it's laid out and almost ready to go to print.
A proofreader is looking for any issues that cropped up through the design/layout process, including text inadvertently left out, unfortunate page breaks, headings not aligned, incorrect images used ... accidental weirdness of any kind. (Ever see a news story about a fatal accident laid out right beside an ad for a day spa whose headline reads "You'll think you died and went to heaven!"?—that's something the proofreader should've flagged.)
In some cases the proofreader is checking that the copy editor's edits were properly captured. And while they're at it, the proofreader is also looking for stray mistakes that all the previous editors missed—typos, mostly, but anything else, too.
(And let's just say it right now: yes, the other previous editors will miss things. We are human. This is why we do multiple passes. It's also why your proofreader should generally be someone new who wasn't involved in any of the previous editorial steps.)