Convert String From Serial Data to Numerical Value

Phew, where to begin...

  1. void move(int x = 0,int y = 0,int s = 0); //Not sure why this needs to be done ???

This is a function prototype. The Arduino IDE handles these for you, so typically these are not necessary.

  1. move(x, y, s); // DO NOT UNDERSTAND "move()"

It's just a call to the function with the provided values.
if( Serial.available())       // if data is available to read, this is how to check the serial part for strings
characters, not strings.
char c= Serial.read(); // the string in serial read() is buffered in to c as characters
the character is read, not the string

if (c == ','){    //look at the incoming chars, if it is a ',' then switch the case
      currentCommand++;  //for some reason currentCommand increments????
    }

If the character is a deliminator (comma in this case) increment the number of commands received.

    else {   //if it is not ',' then store chars in string, keep storing the string until a , is found
      val += c;  // same as (val + c = c)val is the next string + the previous character stored in buffer
      //Serial.println(val);

If it's not a comma, concatenate it onto the String val (note the difference in capitalization, String and string are not the same thing).