Ion Petroianu, VA3NOI
A mini USB cable is what you need, Your Arduino Nano already has a bootloader flashed. You will need a USB to RS232 converter only if you want to flash a bootloader on a new Atmega processor, or if you want to upload a sketch without the bootloader.
As Rob said check the COM port you are connecting to. To do that you can go to Device Manager in Windows and expand "Ports"
You will see a Printer Port and one or more Communication Ports. Most of the time COM1 is used by the system and Arduino is installing another COM Port. Get that number and Start your Arduino IDE. Load the sketch and in Tools/Port select the Port your Arduino is connected. Also check your Board and make sure Arduino Nano is selected.
In the IDE bar press the second button from the top left (with an arrow symbol pointing to the right). Wait for the sketch to compile and upload.
Hopefully you will not get any more errors and your Raduino will start working.