When a bond holder sells the bond to a new buyer between the coupon dates, there will be some interest earned on the bond since the time the last coupon date. This interest is called the accrued interested and must be paid to the seller of the bond. Subsequently, the new owner of the bond will receive all the coupon payments on the bond. The accrued interest will be a fraction of the coupon rate, depending on the date of transaction in the secondary market, which is somewhere between the date of previous coupon disbursal and the subsequent coupon disbursal.
The clean price of the bond is the price excluding the accrued interest.
In US, the popular system is for the bond buyer to pay all the accrued interest to the seller of bond in the secondary market. The amount that the buyer pays to the previous buyer (or the seller in secondary market) is the clean price, or the agreed upon price of the bond plus the accrued interest. Usually, in developed markets, bonds are traded with the next coupon rate attached, which is called as cum-coupon. However, in rare cases, where the bonds are traded without attaching the next coupon rate, this is called as ex-coupon. The total amount paid, including the accrued interest, is the full price (or dirty price) of the bond, which is the clean price + accrued interest.
However, in such cases where bond issuer may have defaulted in payment of interest in past, the bond usually trades at a flat price, without considering the accrued interest.