Overview: There are a few characteristics of lithium batteries and speed controllers which need to be understood in order to match them up so they are compatible with each other.
Voltage: Lithium batteries have battery management system (BMS) boards which control what Voltage the battery will shut down at to prevent them from being over-discharged. This is its lower Voltage limit or discharging cut off Voltage. The BMS also controls at what Voltage the battery will stop charging at which is its upper Voltage limit. These upped and lower Voltage limits are sometimes referred to as the battery's open circuit Voltage range.
For example, if a lithium battery has an open circuit Voltage range of 25-42 Volts. This means that when the battery charger charges the battery up to 42 Volts the BMS will stop the charge and not allow the charger to charge the battery to a higher Voltage. And when the battery is discharged down to 25 Volts, the BMS will shut down the battery's output Voltage so the battery cannot drop below 25 Volts. When this happens the vehicle will shut down and not run under its own power until the battery is recharged.
Most speed controllers have an under Voltage protection circuit which we refer to as its low Voltage cut off level and this is the Voltage level when the controller will shut off power to the motor to protect the battery from being over-discharged. For example, if the controller has a low Voltage cut off level of 31 Volts and the lithium battery has a discharging cut off Voltage level of 25 Volts then the controller will shut down before the battery pack has been used to its full discharge capacity, resulting in a lower amount of ride time or mileage range than the battery is capable of. Ideally the controller's low Voltage cut off level would be the same as or lower then the lithium battery's discharge cut off Voltage.
Amperage: Lithium batteries have a current (Amp) limit where their BMS board will shut down the battery's output power to protect the battery from overheating when too much current is being demanded from it. For example if a lithium battery has a maximum continuous discharge current of 30 Amps and a speed controller demands 31 Amps from it then the lithium battery's BMS board will shut the battery down. Ideally the controller's maximum current rating should be less than the lithium battery's maximum continuous discharge current rating, otherwise the controller could cause the lithium battery to shut down if it is demanding more current than the lithium battery is rated to output.