What unit of measure best describes a dump truck?

It really depends on the commodity being loaded. More dense material (such as asphalt or stone) will be measured by weight (depending on your location, it would either be (US short) tons or (metric) tonnes - seeing as you referred to it as a 'dump truck', rather than a 'tipper', I'm assuming you to be in the US).

For lower density/higher bulk loads, such as fill dirt, wood chips, mulch, etc., the measurement given would be by volume, rather than weight. In the US, these are referred to as yards, for cubic yards.