What is a scientist who uses carbon-14 dating actually measuring?

In all cases the scientist is trying to measure the amount of radioactive decay of carbon-14 isotopes in order to establish the amount of time that has passed since the death of a living thing. As such all methods are based on establishing the remaining carbon-14 in a sample. However the specific type of carbon-14 dating being used will dictate how this is done.

In the traditional approach the scientist is actually measuring the levels of beta activity from a sample. This is given off by the carbon-14 and so the amount of carbon-14 is established as the beta particles are used to identify the ratio of carbon-14 to its non-radioactive carbon-12.

More modern methods use accelerator mass spectrometry and an atom counting approach to actually measure number of atoms of carbon-14 in a sample. This is generally more precise, although often significantly more costly, and both methods are still in use depending on the lab.