Dirty Price is the net present value (NPV) of a bond including all future coupon cash flows PLUS any accrued interest from the last coupon payment date to the next coupon payment date.  Dirty price and clean price are equal on the first day of a coupon period, since no interest has accrued yet on that date.

Dirty price is typically used by banks, funds, and other institutions to calculate daily Profit and Loss for their bond portfolios.  The use of Dirty Price vs. Clean Price has a significant effect on daily P&L from an accounting standpoint.

Bonds are not usually quoted using Dirty Price.  Rather they are quoted using Clean Price (no accruals included).  Taking into account the accrual calculations during partial coupon periods (aka "stubs") complicates the Dirty NPV calculations and results in more NPV volatility.