There are many chapters of bankruptcy, which can lead to a lot of confusion about filing. For the sake of personal bankruptcy filings, there are really only two chapters that are common and relevant for most people: Chapter 7 bankruptcy and Chapter 13 bankruptcy. Most people will end up filing the former rather than the latter, and this is mostly due to the fact that these two chapters have different income requirements.
If the differences between these two forms of bankruptcy could be boiled down to a single statement, it would revolve around the mechanism used to help the bankrupt individual. Chapter 7 uses liquidation to clear out the individual's debts. This means many of your possessions will be sold off to pay your creditors. However, don't despair: some of your most important assets could be exempt from the liquidation process.
For Chapter 13, the mechanism is much different. A repayment plan is negotiated and maintained, allowing the bankrupt person to reorganize their debts and pay back his or her creditors over time. Once the repayment plan is in place, the bankrupt person should do everything in his or her power to follow it. Missing payments or becoming lax with the overall plan can lead to serious consequences.
Each form of bankruptcy has its purpose, and though the differences between Chapter 7 and Chapter 13 are stark, they are still powerful tools in your arsenal to combat debt. If you have questions about either of these forms of personal bankruptcy, consult with an attorney.