If someone on Facebook continually insults you, upsets you, threatens you, or makes unwanted sexual advances, ask them to stop. If they do, fine. If not, you've got a stalker.
First, ask yourself if you feel physically threatened. If the bully is threatening you with violence, or stalking you in the physical world, or coming to your home, the problem is beyond cyberbullying. You need to call the police.
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If things aren't that bad, try to reason with the person. Have you wronged them in some way Is an apology in order See if you can open a real dialogue.
But if that tactic fails on the first try, stop using it. You don't want to appear weak or groveling.
Next, block the jerk so that they can't easily send you anything on Facebook. Here's how:
But blocking them may not be enough. Even if you use Strict Filtering, some messages can get through. According to Facebook, "someone you're not connected to...may pay to ensure their message is routed to your inbox instead of your Other folder." Not a good policy, in my opinion.
In the end, going public may be your best solution. Post an explanation on your timeline, explaining what's going on. Tag mutual friends to make sure they see and read your story. And talk about it, in the real world or by phone, with people close to you. Being bullied is much easier if you have a network of supporting friends.