REO stands for Real Estate Owned and the acronym came about because that is what lenders started calling their departments who handled the properties that became bank-owned properties because they had been foreclosures.
Virtually all of the lenders in America who are investors or servicers of mortgage loans have REO departments and with the record number of foreclosures, they are busier than ever before. Because REO departments are so busy with managing the properties that they own, they don't really want to waste their time doing unnecessary work, so they have a few things that they may require from someone if you want to buy a home owned by the REO department.
For example, it is common for the REO department to require that you become pre-qualified through their mortgage arm (remember, these are banks we are talking about) and that the mortgage arm verify that you can actually qualify to purchase the home.
So if you are shopping for a bank owned property, don't be surprised if the bank that owns the property requires you to meet with one of their loan officers and get pre-qualified.
Are you required to use the bank who currently owns the property as your mortgage company? No. Or at least, you shouldn't be.
But I suppose anything is possible...