"8 LED Fun" for Arduinos explained?
The setup() method run only once because it is used in order to set the direction of the pins (Input or Output).

The oneAfterAnotherNoLoop() method produce only a blink of the LEDs. It turns on the LEDs, wait a little bit, turns off the LEDs and wait another little bit.

The loop() method is an infinite loop that calls the oneAfterAnotherNoLoop() method in order to produce an infinite cycle of blinking.

In your code you initialize the pins every time and this isn't needed. Try this:

void loop() {
  for (int i = 0; i &lt; 8; i++) {       // Initialize pins
    pinMode(ledPins[i], OUTPUT);      // Set pin i as an OUTPUT
  }
  while (1) {                         // Infinite loop
    for (int i = 0; i &lt; 8; i++) {
      digitalWrite(ledPins[i], HIGH); // Drive LED i HIGH
      delay(250);                     // Wait
      digitalWrite(ledPins[i], LOW);  // Drive LED i LOW
      delay(250);                     // Wait
    }
  }
}