Since generally all modern DMMs are capable of measuring negative as well as positive voltage with respect to the COM jack (analog ground), the A/D converter must have a bipolar power supply. To do that, most late-model designs will bias analog ground to some point between the battery negative and battery positive. For example, on the venerable ICL7106, the analog ground is regulated at 2.8V below the battery positive.
Connecting the COM lead (analog ground) to either of the battery terminals will effectively short out the regulator circuitry that maintains the bias. This could easily lead to destructive currents. In voltage-measuring mode, since the positive input jack leads to the ADC input and presents a very high impedance to analog ground, it is generally safe to connect to either battery lead (as noted by Alex above.) Although I would not recommend this as a trouble-shooting technique.
Some older Fluke designs like the 8060A connect the battery negative straight to analog ground, and use a charge pump to generate a negative power supply rail. These units can indeed measure their own battery voltage.
Detecting a low battery condition can be done in all manner of ways, but it isn't done with the DMM's own A/D converter. A comparator referenced to the regulated supply, or sometimes on cheap meters, you may find it's just a voltage divider between raw battery voltage the regulated supply, biasing the B-E junction of a BJT.