A bond premium is extra money a company collects when it issues a bond with a higher interest than the current market rate.
Let’s say my company issues a $100,000 bond with an interest rate of 12%. By the time all the paperwork is completed and the bond is ready for sale, the current market rate is 8%. My company still has to pay 12% because that is what is printed on the bond. An investor likes my 12% rate so much that he pays me $111,000 for the bond. When it comes time to repay the investor, I only have to repay the original $100,000. The rest is a premium for me to keep.