Ignore list is a way to control users you don't like.
Users in your ignore list won't be able to send you private messages and can't reply in your topics.
You will not see topics by ignored users in the digest, Newest Topics, Active Topics.
(Users in your ignore list can reply in other users' topics in which you are also replying, and in such case you'll still see their posts. We consider third-party topics to be neutral territory.)
To add a user to your ignore list, go to their profile and click "Add To Ignore List" link.
You can manage your Ignore List under My Profile.
Your ignore list is private. Other users can't see who among others is on your ignore list. But each user who is on your ignore list will know they are on it.
The option to Extend An Olive Branch (link on the profile page) is a way to restore communications between two users.
If you get into a fight with someone and they add you to their ignore list, then there is no way for you to contact them to cool things off. Your PMs would be blocked, you wouldn't be able to post replies in their topics, and they wouldn't see your topics in the Newest Topics list or in the digest.
One way to resolve the situation is to go to their profile and click Extend An Olive Branch link. This would generate a private message to that user, informing him/her that you are requesting them to give this another try.
At that point, that user might choose to remove you from their ignore list or might choose to keep you on it. It's up to them. But at least, this provides a way to let the other user know that you are willing to stop fighting with them.
The same works in reverse, when you add someone to your ignore list. They can send you such request, and you can decide if you want to remove them from your ignore list or not.
Extending an olive branch to another user automatically removes that user from your ignore list, if he/she is on the list. This is done to prevent "loops" where two users ignore each other and can't reply to each other's olive branch requests because they are on each other's ignore lists.
So when you extend an olive branch, you automatically remove that user from your ignore list. And at this point, it's up to them to do the same or not.