When a dentist accepts assignment of benefits, it means that they will file the insurance claim for you and then the insurance will directly reimburse the office. The patient is then responsible for any outstanding balance.
In offices where they don't accept assignment, it's up to the patient to file the insurance themselves. They are responsible for paying the balance in full for treatment rendered, and then they are reimbursed by their insurance company.
What you decide to do in your future practice is dependent on your area and on personal philosophy. In "wealthier" areas you can get away with not accepting assignment a lot easier, while in an area with lower-income patients you could possibly lose business.
IMHO, as a dentist it is part of my job responsibility to file for my patients. I know it will end up costing me more money and I will need to hire an employee pretty much just to deal with it, but it's a service I plan on providing. Many times patient's can't haggle with ins. cos because they don't know the specifics of treatments, etc.
Of course, I'm sure after a few years of practice this philosophy is subject to change. Dr. Jeff can correct me if any of this is inaccurate.