The word foreclosure is a process and it's the process of putting the property into an auction because the owner defaulted on the loan agreement in some way. A home that's in or going in the process is called a foreclosure. The trustee advertises it and it goes to auction on the courthouse steps for cash. If no one buys it the bank usually takes it back and now it is called a Bank Owned or REO property. A foreclosure is just the wrong term, but everyone seems to incorrectly use it.
REO stands for Real Estate Owned. An REO is a foreclosure property, but not all foreclosure properties are REO properties. When a property goes to the Sheriff Sale, or Trustee sale it is a foreclosure property. If nobody bids on the property, the bank buys it back and it becomes an REO.