The New York Sun has the clearest explanation of covered bond that we've seen.
A covered bond offers the holder what is known as a dual recourse. Normally, a bond is backed either by collateral or by a guarantee from the issuer. A covered bond is backed by both: One issued by Bank of America is backed by mortgages on the bank's balance sheet, as well as by the solvency of the bank itself. If the bank fails, the bond holder has a claim on the mortgages; if the mortgages fail, the holder has a claim on the bank.