Word or phrase to describe how frequently a value is measured
The name for the frequency the samples are taken is Sampling Rate (not specific to audio only). It is measured in samples-per-second or in Hz (which is 1/s). The time between two samples is called sampling period, and is given in units of time. So in your case the period would be 1ms and the frequency 1/1ms=1kHz. Or in terms of rate it's 1000 samples per second.