"Short" means that the anticipated sales price is so low that after deducting closing costs the amount left for the seller is short of their mortgage balance.

So, since the mortgage company must release the lien on the property for you to get your own mortgage and not have to pay back the old mortgage, the shortage amount needs to be forgiven by the lender. This process of short sale approval is started usually prior to seller's offering the house for sale.
He notifies the mortgage company that he can no longer afford to stay in the property and pay the mortgage. He includes a letter describing the nature of his hardship and attaches an opinion of a broker as to what price the property should sell for.

The current lender then places that file in the stack of hundreds of similar files, including foreclosure-bound properties, to answer. Your offer may arrive (probably) before the seller has gotten approval of an amount, and the lender won't usually look at a file unless it has an offer, anyway.

They will receive your offer, agreed to by the seller, and once they begin working on the file, they will order an appraisal or a broker's price opinion to see if your offered price is in line with the market. If so, they will calculate a bottom line price (which normally means they are forgiving thousands of the loan balance), and send an approval letter to the seller and/or the title company where the deal is being closed.

You pay as offered, the mortgage lien is cleared off according to the approval letter, and you're the new owner with your own mortgage to pay.

if you're buying a short sale property, don't expect to close in less than 2 months.