A low-dropout or LDO regulator is a voltage regulator that has a very low input-output voltage drop, thus is able to operate at input voltages very close to the output one. Read more at Wikipedia.
Below is the schematics of a simple LDO regulator with a over-discharge protection circuit for the battery.
The voltage regulation is done with the popular TL431 adjustable shunt regulator, by controlling 3 P-MOS transistors (Q1,Q2 and Q3) placed between the input (the battery) and the output (the load). When the voltage across R1 is above 2.5 V, U1 starts to conduct current through its cathode, increasing |VGS| on Q4 P-MOS. Q4 turns on lowering |VGS| on Q1, Q2 and Q3, which in turn will start to increase its RDS, lowering the output voltage, automatically lowering the voltage across R1.
This negative feedback loop will keep the voltage across R1 at 2.5 V, which means the output voltage will be:
where VREF is TL431 typical voltage reference, 2.495 V.
Things to consider when adjusting the circuit:
Below are two simulations that shows the voltage drop caused by the regulator (excluding VDS on Q6) at 100mA and 1A, by setting the input voltage equal to the regulated one, 3V.
At 100 mA the voltage drop is only 3.38 mV (plus 10.14 mV caused by the over-discharge protection transistor Q6).
At 1 A the voltage drop is 33.1 mV (plus 99.3 mV caused by the over-discharge protection transistor Q6), 10 times more, as expected.
The over-discharge protection circuit cuts off the battery when its voltage drops below a certain limit. This is done by a very low power comparator with an internal voltage reference, LT6700-1. Only one comparator from the package is used, and when the voltage across R7 drops below 400 mV, the output is pulled high turning off Q6, cutting off power to the regulator and load. The cut-off voltage is:
where VTH(F) is the falling input threshold voltage of LT6700 internal hysteresis, 393.5 mV.
When the power is cut off, it is important to keep quiescent current as low as possible to prevent any further battery discharge. In the off state the quiescent current is formed by:
and it totals to 14.9 uA.
For convenience, I've made a small form that lets you determine the output voltage and the cut-off voltage by adjusting specific resistors: