Why do we need mosfet driver like tlp250 to drive mosfet like IRF540 (N-chanel),and transistor to drive relay like jqc-3fc(t73)dc6v?
Why cannot we directly connect and send signals to mosfet and relay?

 

Well first off you like your arduino.
Then comes the fact that if you hook something to a port pin that uses more then 20 to 25 mA you'll be buying a new arduino.
The atmega is easy to pop port pins don't no why they boast about 25 mA a pin because there easy to pop pins
blow regulator kill usb to serial chip on one.

Let's say you hook a motor a little motor the motor starts it pulls 300 to 600mA your arduino is not going to be around long. Same thing with the relay 20 ohm coil that's 250 mA 10 times what the pins rated for.

Now back to the 3fc(t73)dc6v just use a 2n2222 and emf diode. you'll be fine.
I'd use the 2n2222 or if you have to say i used a mosfet I'd use a 2n7000

That relay only needs 60mA to work @ 6 volts

 

MOSFET drivers may or may not be needed depending on the exact situation. If the MOSFET doesn't have a logic level gate, you won't be able to do much controlling it directly with an Arduino. If it does, it becomes more application specific. Slow switching can probably be done directly from Arduino pin to a logic level MOSFET gate, possibly with a low value resistor in between to limit inrush current. Faster switching, like high speed PWM, requires a gate driver to accomplish clean, fast switching. An Arduino output can't provide enough current to overcome the gate capacitance in high speed situations. I'm oversimplifying this description a lot, but this is the general idea.