I can't speak to your situation specifically, as each community has their own policies for appeals. They also have their own political structure and ways of working within that structure to get what you want. I can, however, give you some guidelines that I have utilized in my community in the past.
- Post in complete sentences, using mostly correct English. They probably are not going to mark you off for simple spelling mistakes, but many are not going to read your post if it looks like you typed it from your phone to your teenage friend. Plus, you have more than 140 characters. There is no reason to jam the appeal into a single tweet sized message.
- Stay polite. If you lose your cool, it's much less likely we are going to want to work with you. Remember, this is an online community. Your world isn't going to end if the appeal is not successful.
- Explain why you think your ban was inappropriate. Don't ramble. Don't write a novel. Make this short and to the point.
- Don't lie or make excuses. You've admitted here that you made a mistake. Do the same there. Honesty is going to hurt you a lot less than getting caught in a lie.
- Answer questions from the admins. It's entirely possible you didn't provide a crucial bit of information or they need to wait for portions of their appeals policy to kick in (ie. the banning admin needs to speak up, you need to wait X days after an appeal, etc). If an administrator asks you a question, you should probably answer it. When you do so, follow the first two bullet points above.
It is entirely possible that your appeal may still be denied. Consider your reaction after a denial as well. If you fly off the handle and throw up insults at the admins, the community, and family members of the community, you'd burnt every bridge you have left and then tossed some gasoline on the flames. If, however, you accept the ban and thank them for their time (if you are open to being that considerate) and fade into the internet, there is a chance you may be able to appeal again at a later date.
From my experience, if I forget who you are because you just faded away there is a much higher likelihood you'll be allowed back at some point in the future. If I remember you because you did or said something insulting after a failed appeal, I'm not that inclined to support you returning later.