Short Sales are homes that are for sale but the seller owes more on the bank loan than the property can sell for, hence, it's short of paying off the loan. However, there is nothing short about the process and the time frame to complete the sale. The average short sale takes 6-9 months to complete and often takes longer.
The reason it takes so long is because the seller has to convince the bank to accept the sale, even though the selling/offer price is less than what they owe, and for the bank to write off the difference. Depending on the bank, the amount owed, and other factors, is what makes the bank's approval process a long one.
Sometime the seller has taken the steps to gain bank approval prior to putting the home on the market, this is called "Approved Short sale" or "Bank approved price". This means the bank as agreed to the sale and is typically listed at the price the bank is willing to sell the home for. All short sales require the bank to approve the sale.
If the bank doesn't approve the sell, then they may counter the price with a different price or flat out deny the question and move forward to the foreclosure process.
Short Sales take a lot of patience as it take time and a lot of back and forth between buyer, seller, agents, and the bank; as well as, any lawyers involved in the process. In addition to the time it takes, you will need to be prepared that the slightest delay in returning a document or other factors can completely end the transaction and there is never a guarantee the sale will complete. But the benefit is that you can get an opportunity to buy a home below market value.
Foreclosure homes are homes that the bank has foreclosed on and the bank is selling the home. There are several ways banks sell homes that they have foreclosed on. Sometimes these homes don't ever hit the open market and are sold at auction, often at the county courthouse or exclusive auction sites. Sometimes they will list them in the MLS, in which real estate agents will be allowed to show the homes. (Auction homes, you won't be able to access the home). Banks will not do any repairs and are sold "as is", so there are some risk in buying foreclosed homes, since you won't know what the condition of the homes are. There are some cases where you can do an inspection. There are many different process and rules, so you will definitely want to work with an agent if you plan on buying a foreclosed property.