Just to expand a bit more in detail, let me know if I've left anything out.
What can I edit for the badge?
You can edit any normal questions or answers, including questions or answers which have been converted to community wikis. You can also edit tag wikis and their excerpts, which are both counted separately as edits.
What edits do not count towards the badges?
Edits made on your own posts.
Edits which only change tags.
In order for the edit to count, you must change something in either the body or the title. This can include changes to the tags as well, as long as one of the body or title gets edited with it.
Multiple edits to the same post.
Only the first edit to a single post will count.
Posts which get deleted after your edit.
The edit will only count as long as the post in question remains intact. If at any point it gets deleted, the count with decrement.
Edits made on comments.
Actions taken on the post.
Since you are not actually editing anything, actions such as closing, deleting, rollbacks, and converting to community wiki do not count.
Where can I view the number of edits I've made?
You can view the number of edits that currently count towards the badges in the review panel. Go into any one of the review tasks and hover over the progress bar located at the top and to the left of the tabs. The number at the left of your Copy Editor progress bar is the number of eligible edits you've made so far.
If you suggest edits, do not trust the number of approved suggested edits that are listed on the suggested edit screen. This number is for all the suggested edits you have made, not particularly the ones that count towards the badge(s).
What if my edit gets rolled back?
The edit you made remains completely intact for the post. Since it was an actual, physical edit that could in the future get rolled back too as well, it will still count towards your overall progress. Only the action of rolling back does not count.