There are a lot of different flavors of copy editing. All of them involve taking a very close look a the details of a manuscript and fixing the errors. Here's a quick rundown of some of the types and what I do. Just keep in mind I turn down a lot of copy editing*, especially the proofreading variety because it makes my eyeballs bleed and my soul long for the sweet release of death.
Basic Copy Edit/Proofreading (¢2/word)- Fix the typos and outright errors of formatting, grammar, punctuation, etc.
Regular Copy Edit (¢3/word)- Like a basic CE, but more thorough. I'll mark (but usually not change) language issues.
Heavy Copy Edit (¢3.5/word)- Kind of a hybrid of a regular CE and a line edit. Focused on both prose and errors.
Formatting** (¢50/page)- Standard ms formatting and correcting the little errors you probably don't realize you made.
*If you're looking for a copy editor (especially proofreading), you should probably go to Effie Rose. She's very good, and for some reason she likes that kind of thing.
**I'll usually take formatting work. Something about the manipulation of data scratches a masochistic itch inside my editor's soul.