"Asset" is another term for property. In bankruptcy, anything you own and anything you might have a right to own at a later time is an asset of your bankruptcy estate. Exempt assets are protected in a bankruptcy. Non-exempt assets may be sold (or liquidated, in the case of intangible assets) by the bankruptcy trustee. Each state has its own set of exemptions.
There are 3 different types of asset/property:
Real property: land and structures attached to the land. If you own your home, you own real property (even if you're still paying on the mortgage).
Personal property/tangible property: Anything that you can touch that isn't attached to land. This can be anything from your clothing to your car to the strange looking doll your grandmother gave you when you were a child that's still in your closet.
Intangible property: Property that you have an ownership interest in but you can't touch. Bank accounts, retirement accounts, an interest in a legal action against someone, and the the tax refund you're expecting are all examples of intangible property.