In a Chapter 7 bankruptcy, the individual is allowed to keep certain exempt property. Most liens, however (such as real estate mortgages and security interests for car loans), survive. Many types of unsecured debt are legally discharged by the bankruptcy proceeding, but there are various types of debt that are not discharged in a Chapter 7.
Common exceptions to discharge include child support, income taxes less than 3 years old and property taxes, student loans (unless the debtor prevails in a difficult-to-win adversary proceeding brought to determine the dischargeability of the student loan), and fines and restitution imposed by a court for any crimes committed by the debtor. Spousal Support is likewise not covered by a bankruptcy filing nor are property settlements through divorce. Despite the non-dischargeability of these debts, they must be listed on your bankruptcy schedules, as all debts (and all assets) must be listed.
Chapter 7 or Chapter 13
Individuals can file for bankruptcy in a federal court under Chapter 7 ("straight bankruptcy", or liquidation) or Chapter 13 (a "consumer reorganization", or debt adjustment case). (Although individuals can technically file Chapter 11 bankruptcies, those filings are rare. They typically happen if the individual's debt load is too high for a chapter 13 and they do not qualify for a chapter 7).