Jacob: Someone asked me about this and I remember it was confusing the first time I did it. This is the entire process for reference.

1. Email the professor (optional): It's best to ask the professor if they would approve of you taking the class first, tell them your relevant coursework and grades and anything else you have done to prepare.
2. Submit the form: Fill out the request form on the math website. When you submit it, the form gets sent to the professor of the class.
3. Professor review (1-30 days): The professor will choose to either approve or deny your request
4. Advisor review (1-14 days): The graduate advisor approves or denies the request. In previous years the advisor always approved if the professor did. At this point, you get an email saying whether you were approved or denied for the course.
5. Permission change (0-2 days): Finally, someone changes your permissions in the database. You get a second email verifying you are approved to take the course. It is only after this that you are able to register.
6. Register the same way you would for an undergrad class

Make sure to check in with the professor or advisor if they are taking too long. Sometimes the form mysteriously vanishes partway through.