There are several steps to getting your account unbanned/unmuted. There are also a lot of things to keep in mind. Here's how:
To do this, go to “appeal an offense/ban” on the main runescape page. Log in and it will show your offenses. Each offense will say “appeal” next to it. Do not appeal until you have read this entire guide!
When you click appeal, click “Customer Support has made a mistake” NO MATTER WHAT! Apologizing does not recover your account.
When appealing, you have to make it clear to never break the rules again. (and actually do it too) You also have to make up a story that is not your fault entirely, but is made clear that it will not happen again. Some examples:
“My friend and I were playing, when I had to go to the bathroom. I came back and he was breaking the rules! I am sorry that I trusted him and now I know I cannot leave my account logged on again.”
“My friend was looking through my things, when he found my password written down. He went on my account on my computer when I was not looking. I now know to hide my password in a more secure location and pay more attention when my account is at stake.”
-Keep in mind that Jagex doesn't ALWAYS read the appeals. But you shouldn't take chances. If you really care about your account you should try your best.
I have experienced that the worse your rule, the less likely they accept. For example, if you broke rule 1, they will probably accept. If you broke rule 2 or 3, they will probably deny it. So hopefully you broke minor rules, or you're in big trouble.
I have made this guide because I have had my account banned, and to this day it is still muted. I did not appeal the correct way, and I do not want others to do the same.