The appraisal process is guided by something called the Method Definition Document (MDD). If you are not familiar with the appraisal process already (which you aren't 'cause you're asking the question), you don't want to try to read this unless you're doing some whacky exercise in self-hypnosis.
The MDD is based on a requirements spec called the Appraisal Requirements for CMMI (ARC). And, reading of that document is strictly forbidden without a prescription or special medical dispensation.
The documents are available for download from SEI's Web site and are actually well-written and really useful if you're a Lead Appraiser or studying to be one, but otherwise it's gonna sound like gibberish. Don't say we didn't warn you.
Just so you understand that the complete answer to this question is ordinarily delivered in 2 days' worth of training. We're obviously limited in what we can explain here.
We're going to pick up the appraisal with the portion of the appraisal that most people think about: the on-site period. It's that period of time when there's an appraisal team at your company and they're looking at your evidence and conducting interviews (or performing some other accepted form of verbal affirmation). It's at the end of this period that a company gets the results of the appraisal and, when all goes well, a rating.