Serial.begin () also works without a PC, or not?
Yes, it does.
Because when i unplug arduino from computer and try to read a serial information the LED not want to blink
Which LED? My Arduinos have at least three each.
If you are referring to the RX and TX LEDs, those are controlled by the USB-to-serial converter chip, not the ATMega328 chip. If you are connecting directly from the RX and TX pins to the other chip, the TX and RX LEDs should not be expected to flash.