How is a Dragonball character's power level determined numerically? What exactly is measured?
*This particular image depicts a moment where Vegeta vociferously reacts to the scanner's readout of Goku's power level being above the 9000 measurement.
Power Level is a reading of a persons current Ki.
The actual term "Power Level" usually refers to readings produced by scouters, which are sensory devices used by Frieza's crew that detect a person's ki. The reading is converted into numbers of an unspecified unit, and the resulting number is the "Power Level", or an estimation of that person's physical capability.
It is first introduced in Dragon Ball, where a young Goku learns to sense ki after drinking the Ultra Divine Water, although the Z Fighters are able to detect power levels via the ki sensing ability eventually.