First of all make sure that you have programmed your arduino well and that you have connected everything as instructed.
Try putting the line digitalWrite(13,HIGH); at the end of your initialization method just to see if the onboard LED lights up meaning that the program was uploaded correctly and is running.
Second make sure you know which Serial port you're connecting to. If you have onboard serial ports you may have COM0 taken by the "real" serial port and the virtual serial port used by the arduino is something else.
You probably have something like this in the setup() method in processing:
String portName = Serial.list(); myPort = new Serial(this, portName, 115200);
Serial.list() actually returns an array of strings with the names of your serial ports. In my example I'm using the first one in the list (i.e. the one at position 0), since that's the one my arduino uses. But you may try setting that value to 1 or maybe 2 depending on how many serial ports you have (bluetooth for example can create its own serial ports).
I suggest doing println(Serial.list().length); at the beginning so the program will print how many serial ports it finds (so you know how many different ones you can try, eg if it prints 3, you can try any one from 0 to 2).
Experiment around with that. If it doesn't work then I'd check that you are setting the same baud rate on both the processing program and the arduino. That 115200 up there in my example is the baud rate. In the arduino program i have a similar Serial.begin(115200);
The number in both places has to be the same otherwise you'll get corrupt data.
That's as far as I can help for now... try these tips out, and if it still doesn't work send a reply here and i'll try to help you further (I'll build the circuit myself and try it out). However it's usually the simple things that you forget to check first... as i said if you still have a problem... just post again... preferably post some of your code so I can better understand what you are doing.