Turn on each of the LEDs in order, by setting the cycle delay to 100 milliseconds
I see in your second diagram that you are controlling the ground side of the LEDs. digitalWrite(PinNumber, HIGH) will turn an LED OFF, digitalWrite(PinNumber, LOW) will turn an LED ON. Using delay() in a sketch should be avoided.

void setup(){
  pinMode(13, OUTPUT);
  pinMode(12, OUTPUT);
  pinMode(9, OUTPUT);
  pinMode(6, OUTPUT); 
  digitalWrite(13, HIGH);
  digitalWrite(12, HIGH);
  digitalWrite(9, HIGH);
  digitalWrite(6, HIGH);
}

void loop(){
  digitalWrite(13, LOW);
  delay(100);
  digitalWrite(12, LOW);
  delay(100);
  digitalWrite(9, LOW);
  delay(100);
  digitalWrite(6, LOW);
  delay(100);
  digitalWrite(13, HIGH);
  digitalWrite(12, HIGH);
  digitalWrite(9, HIGH);
  digitalWrite(6, HIGH);
  delay(100);
}