A short sale is where the sales price of the property is less than what the seller still owes and the lender is willing to accept less than they are owed.
Why would a lender agree to a short sale? Mortgage lenders are in the business of making loans not owning property. When a loan is in default, it is viewed by the lender as a non-performing loan. In addition to not earning interest on their loan the Federal Reserve requires the lender to put aside funds to cover the bad debt. These funds are called a reserve and cannot be lent to other clients. In addition there are some rules about how many non-performing loans can be kept on the books and the punishments for exceeding these limits are serious so banks and other lenders are anxious to get these loans resolved. It should also be noted the foreclosure process is long and expensive for lenders. The reason many investors look for bank owned properties is, historically banks have not been very good at selling property. All of these reasons are why lenders are willing, in some cases, to take less than they are owed.