"Paying your dues" might be defined as doing work that may be considered grueling and demeaning because you are the newest or youngest person on staff. Or, it could involve working longer hours with less time off or taking less desirable assignments so that you can gain seniority. If you're a recent graduate, your company probably sold you on contributing your skill set in a meaningful way, and you may be insulted that your pricey college education has bought you a position answering phones or other jobs that might be expected as an "assistant."
Although the tide is changing and mandatory dues-paying is not as rampant as it once was, don't expect to be running the company within days of your start date. Think of your time at the entry level as a rite of passage. Serve with a smile, and attitude and show you're eager to learn everything you can about your job, your organization, and your field. Once you've spent a year in the trenches, you will appreciate the basic tasks that go into running a business, and you will also have the knowledge and experience to contribute in a consequential way.
How long should you be expected to pay your dues? Some might argue that work is a permanent membership and as such, we pay dues forever. But one year is about the length of time you should spend as an assistant. Set clear promotion goals for your boss to ensure that this is the case, and take on projects above and beyond your daily job responsibilities to show what you're capable of.
If you're doing your job well and there is no opportunity for advancement in sight after your anniversary has passed, update your resume and start looking. Some managers will keep promising twenty-somethings in administrative positions out of need (i.e. no one can do your job as well as you can) or jealousy. Move on before you get stuck, and think carefully before accepting your next position. A series of lower-level jobs held over long periods of time can impact your future employability, so check to see if other people at your level have had the opportunity to move up after a reasonable period of time.