Firstly, you'll need to have some information handy. You'll need the name of your account, the email address connected to that account, the answer to your secret question (if you can remember it), the names of some of the people you had in your friends-list, and if you had payment information on file, you'll want to have that information handy.
Essentially, you're going to have to start by proving that you are actually the person to whom that account belongs.
Second, you will need to contact Linden Lab, in one of three ways. By post, by fax, or by submitting a support ticket.
Second Life Support
945 Battery Street
San Francisco, CA 94111
+1 415 651 9221
Go to http://secondlife.com/support and submit a ticket as 'a guest', using your email address (because, being banned, you probably cannot use your Second Life account login).
For Ticket Type, select Special Questions – Basic account or Guest Login.
For the next field (which appears once you have selected that), select Account Issues.
Under Basic Account Issues, select The System says my account is Disabled or Suspended.
Now, the next field that appears will be Disabled Issues. When your account was banned, Linden Lab sends an email to your specified contact email address. If that address isn't working, or you're behind one of the more aggressive spam filters (like the one that Earthlink uses), you won't have gotten that email.
In that case you'll need to select I do not know why my account is disabled or suspended. Otherwise, you're probably going to need I believe my account was disabled for a Governance issue.
What do I say?
Now, whether you're posting in your request, faxing it or using the ticket system, a key thing to remember is to be clear, precise, on-topic and brief.
You may just have been unlucky, but you might well have done something wrong. Either way you need to explain what happened. If you were banned because you broke the rules when retaliating against someone who griefed you, or because you misunderstood or were deceived about the rules, explain that. Breaking the rules is a banning offense, regardless of the why you did it, but you can still appeal and attempt to get your account restored.
It's possible that you did something with a hacked viewer like Shooped Life – lots of people who use that one get banned. Some of its tools trigger all sorts of alarms.
Linden Lab has access to the logs, so they know who did what, who said what, and when, so you can't lie to them about any of that. If you try, your appeal will automatically fail. The logs are a bit tangled, however, and they can't tell what you were thinking or intending at the time.
You get exactly one chance to get your Second Life account unbanned, so don't waste it.
Explain what happened, what you did or said and why. Keep to the point. Proofread what you wrote, and maybe get the spelling checked. It seems like a minor detail, but it's worth doing.
If you did the wrong thing, own up to it and apologize. If you did the wrong thing and Linden Lab doesn't think you learned your lesson, you've blown your chance. We all make mistakes sometimes, however, and this is your one and only shot to show that you've learned from it.
Do not at any point attempt to argue that the rules are dumb, or that the rules shouldn't apply to you, and don't be rude. Linden Lab isn't interested in your opinion on that matter. They're only interested in knowing that you're not going to ever break the rules again.
Don't write an essay. The people reviewing your appeal would far rather read one page than thirty. Don't add anything unnecessary. Don't insult, disparage or write rude things about other users. Stick to the facts.
What happens next?
If the team reviewing your appeal needs more information from you to establish your identity or about the events that led to your banning, they'll ask. You'll need to be prepared to send them anything they need, possibly including copies of some of your identity documents.
Appeals take longer than regular support tickets. You wouldn't want someone just glancing at your appeal for ten seconds and then throwing it away, now would you? So, there's usually a bit of a queue. Linden Lab tries to respond within three working days, but if you haven't heard back in four weeks, you should consider trying again. Your account and inventory are usually held for several months just in case your ban is reversed.
Linden Lab may choose not to unban your account. It happens. Unfortunately, that's life. If they decide that they will not, they won't tell you why they will not. Likewise, at no time will Linden Lab tell you who reported you, or violate the privacy of any other user (banned or not). Exactly how each appeal is reviewed is also kept intentionally private, but Linden Lab does assure us that each appeal is given proper consideration.
What, exactly, that consideration consists of depends entirely on you.