When filing for Bankruptcy certain property of the debtor is exempt from the Bankruptcy estate, which means that it is not subject to being taken by the Trustee and used to pay your debts. In simpler language, exempt property is property you get to keep.
When filing a Bankruptcy as a resident of Massachusetts you can choose to use the exemptions allowed under either State or Federal law, but you must choose one or the other.
There are many exemptions that are similar under both schemes, such as the exemption of most qualified retirement plans. The most significant difference is that under the Federal exemptions you are only allowed to exempt approximately $20,200 in primary residence equity, while under the Massachusetts exemptions you can exempt up to $500,000 in primary residence equity (assuming you have filed a valid Declaration of Homestead). This can be of significant importance if you are trying to stay in your home.
The exact amounts of exemptions are subject to change and you should consult with an attorney to obtain the most current figures and to decide which option you should choose.
UPDATE: The Massachusetts Exemptions have recently changed. For more information read our post on the changes: New Massachusetts Property Exemptions: The Return of 2 Cows, 12 Sheep and 2 Swine.