Clean Price is the net present value (NPV) of a bond including all future cash flows, but without including any accrued interest from the last coupon payment date to the next coupon payment date.  Clean price is equivalent to the bond NPV on the first day of a coupon period, since no interest has accrued yet on that date.

Bonds are typically quoted using Clean Price, since this simplified the NPV calculations and results in less NPV volatility.