A foreclosure is a home sold at a courthouse auction, no set price just what the highest bidder is willing to pay. Catch here is the purchase is all cash, with no title insurance, most times sight unseen.
If a lender takes ownership at an auction, it becomes an REO, Real Estate Owned. They usually list it at a slight discount due to condition or lack of warranty. You may hear this was a "foreclosure" home.
A short sale is a bit of a gamble. It is where a homeowner owes more than the home is worth, and has a hardship. They list the home at a guesstimate price and when a potential buyer signs a contract to buy, it is subject to third party approval. That means lender will negotiate a settlement with the homeowner. The buyer has no rights except to be first in line, to decide if they will buy the home at whatever price the lender ultimately agrees to release it at. Buyer is taking home in As Is condition, and lender can actually withdraw their offer at any time before closing. With home values increasing, they are expecting price closer to market value, revisiting home value right before closing. Buyer has no recourse other than to walk away. Many times a short sale takes more than one buyer before it actually sells. This is for someone that does not need a home in a specific time frame.
Sometimes you can find a short sale at "Approved Price", that means lender has reached a settlement price, and might close sooner.