The meter displays the voltage of the 12V system, in some form or another. When the converter is running, this is the converter output voltage; when it's not running because you're disconnected from shore power, it would be the battery voltage. To be more precise, it's really the same voltage in any case; neglecting any voltage drops in the wiring the voltage at the battery and at the converter and at the distribution panel is one and the same voltage. (Under relatively high currents, the voltage drops due to wiring resistance probably shouldn't be neglected, and the voltages may vary measurably.)
Is this a little bargraph display with about three or four lights for voltage? If so, the transition between the lights is probably somewhere between 11 and 12V if it's marked at those voltages, but who knows exactly what (without some testing). Those little monitors are not very useful for determining battery status...perhaps better than absolutely nothing at all, but that's about all that can be said. Some do have some means for "calibrating" them by adjusting a little trimpot, which might make it slightly more useful.
Lead acid batteries don't like being discharged too much. A battery at 12V (resting) is due to be recharged. The lower the voltage goes, and the longer the battery sits discharged, the more likely and more severe damage to the battery can be.