The first piece of advice is to always sign up for classes before the registrar's deadline. Second, as soon as you know you would like to be an Economics major, please declare online. Even with these precautions, you may find yourself bumped by the registrar from an Economics class that you registered for. The first thing to realize is that you may not need to take that exact course in that exact term. Go ahead and fulfill a distributive this term, and try for that Econ class next term. If you are determined to try and get in this term, though, there are several steps you must take.
First, as soon as you discover you have not gotten into your desired course, fill out the Online Waitlist Form. Do not contact the professor of the course. The Department Administrator is in charge of all wait list issues and will contact you about your status on the wait list.
Second, when the term starts, be sure to attend the first class meeting of your desired course. Your attendance will both ensure that you are not behind if you get off the wait list, and it will help gauge whether there will be room in the classroom for wait-listed students. That is, if registered students who are planning to drop do not attend classes and those hoping to add do, the classroom will be at normal capacity and things will look good for moving off the wait list. If the classroom is overflowing, odds are not so good. Thus, it is important to continue to attend your alternate course while the wait list situation is sorted out. Typically, by the end of the first full week of classes (i.e., after three class meetings), you will either have been added to the course or informed that no more students will be added from the wait list. Finally, if you got into the course of your choice, but would prefer to be in a different section, you should follow the same steps. Essentially, you are putting yourself on the wait list for the desired section.