How can use a separate power source for powering Arduino, and another battery for powering the motors? How can I make connections through the motor driver module L298N? I know how normally we use one battery to power the Arduino as well as the motors. But I am facing the problem while using different power sources.
That motor driver has an optional on-board 5V regulator. It can be enabled / disabled with a jumper.
With that regulator enabled you just need to connect a ground and your control signals (ENA/IN1/IN2 or ENB/IN3/IN4).
If you disable that regulator (remove the jumper) then you need to provide power from the Arduino's 5V pin to the +5V screw terminal.
The ENx pin(s) (after removing any jumpers that may be on them) connect to any PWM capable pin (this is the speed control pin). IN1/IN2 or IN3/IN4 connect to any IO pin since these are just used to set the direction or control the braking modes. ENA controls the speed of motor 1, and IN1/IN2 control the direction of motor 1. ENB and IN3/IN4 are likewise for motor 2.