The main difference between these two kinds of bankruptcy is how your debt is discharged. In a Chapter 13 bankruptcy, you restructure and reduce the total amount of your debt; however, you make payments on your debts over a three to five year period according to a Chapter 13 plan. At the end of the three to five year period, you are granted a discharge of your debts.
Chapter 7 bankruptcies are more common, though. In a Chapter 7 bankruptcy, a trustee liquidates your assets in exchange for a discharge of your debts. In this type of bankruptcy, you do not make any payments over any period. Rather, you must give the trustee all of your non-exempt assets (for most people, all of your assets are exempt and you get to keep all of your assets) in exchange for the discharge of your debts (and again, you do not need to make any ongoing payments).