In terms of effect rather than mechanism, a clean ground has a low amount of stray voltage at power line frequency and above, even into the RF spectrum. It can achieve this by being well and truly grounded and bonded and by eliminating sources of current from other equipment that causes voltage on the wires both by IR drop and by electromagnetic induction and capacitive coupling.
A dirty ground is subject to unwanted voltages to remote earth caused both by other equipment and the equipment itself. These voltages can be coupled into sensitive inputs on the affected equipment, often because of poor design of signal and power ground handling in the equipment itself or attached equipment.
An alternative way to mitigate the problems purportedly caused by dirty grounds is to use differential drivers and receivers with good common mode rejection at all signal inputs and outputs and to provide isolation (as with optical couplers) that prevents unwanted currents from flowing on data ground and shield conductors.