Lipo cells have both a minimum and maximum voltage you need to stay within. Going below the min voltage can damage the cell and lead to swelling where the cell literally swells up like its filled with air. At that stage, the cell is damaged and beyond repair and your lipo is pretty much finished as its unsafe to charge or run in the swollen (usually called puffed) condition.
Min voltage is actually 2.7v/c. We typically have a low voltage cutoff (lvc) in the speed control to shut down or pulse the power to let us know voltage is getting low at anywhere from 3 v/c and up. I personally have mine set at either 3.1 or 3.2v/cell to give me a little safety margin. Max cell voltage is handled by your charger and should not exceed 4.2v/cell. Lipo requires a lipo charging protocol. CC/CV, where in the initial part of charge, lipo is charged at constant current (CC) until the max cell voltage of 4.2v/cell is reached, then the charger changes to constant voltage where the current decreases while the lipo is topped off.
There is a diff in tolerance of overdischarge based on cell quality. Higher quality cells will tolerate mild overdischarge better than lower quality cells will - overdischarging a high quality cell and a lower quality cell to the same extent and you'll usually see the lower quality cell fail (puff) while the higher quality cell can or will recover. The higher quality cell will also be damaged, but the damage will show up as a decrease in life (number of discharge/charge cycles) rather than puffing. Higher quality cells will also have less tendency for the cells to become unbalanced (cell voltages not equal) while lower quality cells will show more of a tendency to get out of balance.
This will give you a little background - read the lipo parts