Help for Newbe

The pinMode(pin, OUTPUT) sets the pin to output and the default state is LOW. None of the LED pins are HIGH or ever set to HIGH so nothing can happen.

How is your switch wired? Does it have a pulldown resistor?

Please read the "how to use the forum" stickies to see how to properly post code. Here is your code posted properly:

// Constants

const int ledPin9 = 9;
const int ledPin10 = 10;
const int ledPin11=11;
const int ledPin12=12;
const int buttonPin=2;

// Variables
int buttonState=0;
int ledPin9State=1;
int ledPin10State=0;
int ledPin11State=0;
int ledPin12State=0;

void setup() {  
 pinMode(ledPin9, OUTPUT);
 pinMode(ledPin10, OUTPUT);
 pinMode(ledPin11, OUTPUT);
 pinMode(ledPin12, OUTPUT);
 pinMode(buttonPin, INPUT);
 
 
}

void loop() {
 buttonState = digitalRead(buttonPin);
 ledPin9State = digitalRead(ledPin9);
 ledPin10State = digitalRead(ledPin10);
 ledPin10State = digitalRead(ledPin11);
 ledPin10State = digitalRead(ledPin12);
 
 if (buttonState == HIGH && ledPin9State == HIGH) {
 digitalWrite(ledPin9, LOW);
 digitalWrite(ledPin10, HIGH);
 }
 
 if (buttonState == HIGH && ledPin10State == HIGH) {
 digitalWrite(ledPin10, LOW);
 digitalWrite(ledPin11, HIGH);
 }
 
 if (buttonState == HIGH && ledPin11State == HIGH) {
 digitalWrite(ledPin11, LOW);
 digitalWrite(ledPin12, HIGH);
 }
 
 if (buttonState == HIGH && ledPin12State == HIGH) {
 digitalWrite(ledPin12, LOW);
 digitalWrite(ledPin9, HIGH);
 }
 
}