First I draw a picture as shown below. The drawing does not include the 10kohm resistor used with the switch.

Then I connected the circuits. Looking pretty neat, am I right?

Then I wrote the program. For more details read the comments.
// Current button press
int button;
// Previous button press
int oldButton;
// Number of presses
int i = 0;
void setup() {
//Button circut input
pinMode(18, INPUT);
//LED 1 circut output
pinMode(22, OUTPUT);
// LED 2 circut output
pinMode(21, OUTPUT);
Serial.begin(9600);
}
void loop() {
oldButton = button;
// Read button press HIGH or LOW indicating whether button has been pressed
button = digitalRead(18);
// Just printing the output
Serial.println(button);
// If button has been pressed and previous button press was not active increase index
if(button == HIGH && oldButton != HIGH) {
i++;
}
// If index out of bounds make it zero
if (i > 2) {
i = 0;
}
if (i == 1) {
// If button pressed once light first LED
digitalWrite(22, HIGH);
digitalWrite(21, LOW);
} else if (i == 2) {
// If button pressed twice light second LED
digitalWrite(22, LOW);
digitalWrite(21, HIGH);
} else {
// If button pressed zero or three times turn off lights
digitalWrite(22, LOW);
digitalWrite(21, LOW);
}
// Little delay
delay(10);
}