The answer to your question is a little complicated but I will try to explain to you the different approaches an owner that is in financial trouble can take. When the owner of a property falls into financial trouble and can not afford to make the mortgage payments, there are a few solutions.
First. The Owner can contact his/her Bank and explain the situation and try to work out a Loan Modification. A loan modification, if accepted by the Bank can reduce the mortgage payment to levels that the owner can afford. Some lenders are even bringing down the amount owed to them to reflect the present deflated values of homes in many areas and that way, avoiding to accept short sales or go into a foreclosure.
Second: If the Owner of the home can not afford to make any mortgage payments at all because of job loss, health problems and his/her financial situation is precarious, then he/she could opt for two other solutions.
A) Talk to the Bank, explain the situation and try to convince them to accept a "short sale". In a short sale, the Bank agrees to accept a lower amount than what is owed to them. To my view, this is a good will attempt by the Owner of the property talking to the Bank and trying to negotiate a short sale instead of simply not communicating and letting them foreclose on the Home. In a short sale the Owner of a home has to prove to the Bank that he/she can not afford to keep the home because their financial situation deteriorated to the point that he/she have only a limited amount of income to cover their basic expenses. If the Bank agrees to a short sale, the property comes out to the market as a "short sale". This means that the home will be sold for less than what is owed to the Bank and the Bank agrees to take a loss for the difference. The Bank always has the last word. The Owner might agree on a selling price but the Bank has to accept the propossal. If the Bank does not accept, then, the transaction does not goes through. Some of these short sales end in foreclosures many times because some Banks have unrealistic expectations as to what the home could sell for. In a short sale, the Owner has his/her name on the Deed until the home is sold and they transfer title. Short sales can take from a few weeks to be accepted to many months and many times are not accepted at all, so the properties go to foreclosure. Banks are overloaded with short sales.
B) Let the property go into foreclosure. Usually the Owner of the home does not communicate with the Bank and the Bank starts foreclosure proceedings. In California within 3 months of filing a notice of default , the debt can still be redeemed. After three months went by, there is a 21 day redemption period in which, if the Owner wants to get the home back he needs to pay the Bank the full amount of the Mortgage, missed back payments and foreclosure charges. If the Owner does not respond, the property is sold at auction at the steps of the Court House. If there is no bidder, then the property reverts to the Bank and the Bank becomes the legal owner of the home. The deed is registered and recorded at the name of the Bank.
Usually, Bank owned homes are given to Real Estate Agents to put them out for sale. These agents are well known individuals that the Bank has used before. Bank owned properties are usually advertised as Bank owned property, corporate sale or foreclosed home. In a Bank sale, the property comes out to the market at, usually, a little lower price than the market value. If the home is located in a desirable area, sometimes the price of the home gets overbidded and the home sells for more than the asking price. Offers on Bank sales, opposite to short sales can be accepted right away and can close very fast if the home is vacant and the Buyer can perform in a short time.
I hope you understand this explanation. Otherwise, you can go to many sites that explain this same process in more detail.