For storage charge, it charges or discharges the pack to about 3.8 to 3.9v/cell as it's the safest voltage for storing. If the battery is below 3.8v/cell it will charge / balance it back up. If it's above 3.8v/cell it will discharge and also balance out the cells while doing so. Most chargers have settings where you can change the voltage for it to stop charging or discharge too.
Note that most chargers can only discharge batteries around 1 to 2 amps if you don't have a resistor or discharge bank as it can only take in so much from the battery. Now with charging, it's the opposite so you can put in how ever much you want into the battery (charge rate) most likely 1C. 1C : 5000mah=5.0A charge rate, 3000mah=3.0A charge rate, ect. I charge my packs at 10A on balance charge so IRs can stay low and do it can be charged fully in a faster time for the next race. I discharge in the car. After a race (5 minutes) the battery is already at 3.9 / 3.8v/cell so I put it on storage charge after the race night to balance out the cells.
Now for discharging or running the battery down in the car, most ESCs today have LVC / LVD meaning Low Voltage Cutoff/Detection. You can change it in the ESC settings. I personally like to have mine set on auto or 3.4v/cell. After the battery is discharged more than 3.0v/cell, it doesn't hold a charge and it dies pretty quickly and is unsafe, but it can still be recoverable. Most batteries if ran down below that threshold, if you leave the battery sit, the voltage will rise by its self to around 3.3v/cell.
Discharging can also mean to get the battery completely dead to dispose/toss it.