The terms "foreclosure," "bank-owned property" and "REO" are used interchangeably these days.
Technically, "foreclosure" is a process, not an end result. It's the process the bank/lender goes through once the borrower fails to continue making payments on their mortgage. Eventually, the home is "foreclosed" on and the borrowers are evicted. (That's an "in a nutshell" explanation)
"Bank-owned property" and "REO" mean that the foreclosure process has been completed, the home went to auction and that the bank has bought it. The bank now owns it and when you see it, it's after the bank has chosen a listing broker/agent to help sell it and the property is "on the market."
Short-sales are a different ball of wax. Follow the link below to my post on what a short-sale is.