For one of my books, I was asked to cut a whole set of characters who played a recurring, rather than continuous, role. I was really resistant at first, but when I thought about it, I realized that this actually strengthened the book, because these characters had been playing a sort of deus-ex-machina role in the plot, and without them the plot took on a more organic quality. This was great editing, and I'm grateful for it.
For another book, I was asked to get rid of a character who played a small but pivotal role in the hero's personal journey. I thought about it a lot and decided this was a change I couldn't live with. I explained my reasoning to the editor, and she acknowledged that my reasoning made sense. The character stayed, though I did trim his scenes a bit.
In other cases, I've capitulated on some changes that, left to my own devices, I wouldn't have made but didn't feel were a major issue, and thus gained some leverage for refusing to make other changes I felt strongly weren't a good idea.
Editing always involves change, and is (ideally--not every situation is ideal) a collaborative process. Your editor is trying to make your book stronger, and may have a better perspective on some things than you do. On the other hand, if you feel strongly that a change isn't right, you can refuse to make it as long as you explain your reasoning fully.
- Victoria