In a foreclosure sale, the lender sells the property at auction and the net proceeds are applied to satisfy the unpaid balance of the loan. Any excess goes to the borrower. Generally, any shortfall is forgiven. There are some exceptions, such as judicial sales.
In a short sale, the property is sold on the open market. The lender agrees to accept less than full payment. The lender releases his security interest so that the sale can go through. The unpaid portion of the loan may or may not be forgiven depending on several factors, such as state law and the agreement.
Forgiveness of any unpaid balance, in both foreclosure and short sales, can result in tax consequences. The rules are complicated. Consult with your tax attorney before any action occurs, if at all possible.
Many dollars can be saved with the proper legal advice before a foreclosure or short sale is commenced or completed.