What is a Short Sale?
In real estate, a short sale is when a bank or mortgage lender agrees to discount a loan balance due to an economic hardship on the part of the mortgagor. The home owner/debtor sells the mortgaged property for less than the outstanding balance of the loan, and turns over the proceeds of the sale to the lender in full satisfaction of the debt. In such instances, the lender would have the right to approve or disapprove of a proposed sale.
Extenuating circumstances influence whether or not banks will discount a loan balance. These circumstances are usually related to the current real estate market climate and the individual borrower's financial situation.
A short sale typically is executed to prevent a home foreclosure. Often a bank will choose to allow a short sale if they believe that it will result in a smaller financial loss than foreclosing. For the home owner, the advantages include avoidance of having a foreclosure on their credit history. Additionally, a short sale is typically faster and less expensive than a foreclosure. 
In short, a short sale is nothing more than negotiating with lien holders a payoff for less than what they are owed, or rather a sale of a debt, generally on a piece of real estate, short of the full debt amount.
Lenders have a department (typically called a loss mitigation department) which processes potential short sale transactions. Typically, lenders do not accept short sale offers or requests for short sales until a Notice of Default has been issued or recorded with the locality where the property is located. Lenders have to approve of any buyer's or listing agent's commission in advance, a primary reason for non-brokered short sales with a specialist or facilitator to save on the margin.  Many of these facilitators work with a private lending party for their financing, such as a partner or syndicate.
Lenders have a varying tolerance for short sales and mitigated losses. The majority of lenders have a pre-determined criteria for such transactions. Other distressed lenders may allow any reasonable offer subject to a loss mitigator's approval. "Red tape" is very common in short sales, similar to REO and HUD properties, requiring potentially multiple levels of approvals and conditions. Junior liens, such as second morgagees, HELOC lenders, and HOA (special assessment liens), may need to approve of the short sale. Frequent objectors to short sales include tax lieners (income, estate or corporate franchise tax - as opposed to real property taxes, which have priority even unrecorded) and mechanic's lien holders. It is possible for junior lien holders to prevent the short sale.
While it is frequent if not common for a lender to forgive the balance of the loan in question, it is unlikely that a lien holder that is not a mortgagee will forgive any of their balance. Further, it is common for a lender to omit updating the zero balance and settlement option on the mortgagor's credit report, or even flat refuse to do so "due to their financial loss."