More pedantically, concentration-sensitive detectors respond to the *concentration* of analyte in the mobile phase at any given time (as the name implies). Thus, they tend to be flow-rate dependent (imagine shutting off the mobile phase flow while a peak is in the flow cell of a UV detector; the analyte is still in the cell, so the signal stays high).
Mass-sensitive detectors respond to the *mass* of analyte that has gone through the detector (again, as the name implies

). They tend to be flow-rate independent (imagine shutting down the mobile phase flow while a peak is eluting into an evaporative detector. Once you have evaporated the droplets already in the detector -- i.e., almost instantly -- the signal will drop back to zero).
Either kind may have response that varies with analyte structure.
To expand on DR's post, concentration-sensitive detectors include:
- UV-VIS
- fluorescence
- RI
- amperometric (pretty much!)
- electrical conductivity
Mass-sensitive detectors include:
- MS
- the various evaporative detectors
- coulometric
- any of the old "transfer" detectors (moving-belt FID, etc.)
- post-column reaction detectors (usually)