If you purchase a bond on May 31, and the next scheduled interest payment is June 30, you fall between interest payment dates and you have to pay the seller the portion of the next interest payment he is due, based on how many days into the payment period the seller owns the bond, say 60 days. This amount is called accrued interest and is included in the sales price.
At the end of the tax year, the accrued interest is usually included in the Form 1099-INT that the buyer receives. For the buyer, this amount is not taxable, instead it is treated as a return of capital investments, which reduces his tax cost of the bond. The accrued interest is taxable to the seller. The seller must report the part of the sales price which represents accrued interest as ordinary interest income, even though a Form 1099-INT is not received.