I have to connect arduino with the USB cable right
Yes. The Arduino then appears to the PC as a COM port (also known as a serial port).
is it just for write the code in the microprocessor
The Arduino IDE writes the code to the microprocessor using the COM port.
I don't have to use a RS232 port on my pc or the arduino RX and TX, right?
Right. And, it's a good idea not to use RS232 to talk to the Arduino, since RS232 signals are +12V and 012V, rather than 0 and 5V that the Arduino can tolerate.
There are chips (MAX232) that can manipulate the RS232 voltages to TTL levels that the Arduino uses (and vice versa) if you have some device that only outputs RS232 signals that you want to connect to the Arduino.