Exempt property is property that can't be touched by a bankruptcy, which allows you to keep it and rebuild your life from an established point. So, that means that if you file for bankruptcy and have to liquidate your assets, there are some assets you'll be able to keep and protect in order to prevent you from being without anything you've worked toward.
The U.S. Government has established U.S. Bankruptcy Courts that handle bankruptcy and create a bankruptcy estate for each person who enters it. At that point, a trustee helps liquidate the assets for the estate, allowing money to be raised to pay off creditors.
Not all things that you own will be taken during a bankruptcy. With exemptions, you can keep some of your property after you go through the bankruptcy proceedings. Bankruptcy isn't designed to make you start over, it's designed to give you a fresh start. Things like your primary vehicle or home could be protected with an exemption. Some kinds of exempt property include necessary household furniture or goods, necessary clothing, motor vehicles (up to a certain price), and some of your earned wages that have not yet been paid.
Other property, which is not exempt, will need to be turned over to the bankruptcy estate and trustee. Expensive musical instruments will need to be given to the trustee, unless you're a musician professionally. Any kind of family heirlooms, cash, bonds, investments, or collections of coins or stamps must be turned over as well. If you own a second home or multiple vehicles, any that aren't protected by being exempt will be sold to help you pay off your creditors.
Source: FindLaw, "Exempt vs. Non-exempt Property Under Chapter 7," accessed Aug. 11, 2015