Copy, or line editing, is exactly as it appears. The editor goes line by line and makes corrections to the fabric of your story. They hunt for common errors, punctuation and clarification of flow and generally buff the story. It’s important to find one that you get on with, and is priced reasonably – most editors charge by specifically formatted page size, word count, or less commonly, an hourly rate. You should always ask for a free sample of your first (x) pages (being based on what the editor agrees on) and compare them. As a team, we suggest getting the same pages as a sample, so you can see what’s coming back from everyone, and what’s actually being missed. Choosing from there should be easier.
An analogy of precious stones
If you think of your story as an uncut diamond, you make the general cuts, your story editor can check them, your line editor makes the biggest refinements, and your proofreader buffs everything out.