Got a denial letter for unemployment, what now?

My claim was denied after 12 weeks of waiting, and the letter says I have a short window to appeal. My employer's version of events does not match mine, and I have emails that show it. If you have appealed before, what helped you prepare?

Thanks for writing this out. I went through something close to it last year, and it settled down once I kept a simple log of every call and form.