Basic Arduino Question. Trying to turn off leds turned on through the serial monitor by using a push button
To turn off all leds when button is pressed, you must detect a rising or falling in the button signal. 



In your code you detect the change only if serial is available, but you should do it in the loop.

The simple way to detect the change is doing this in every loop:

if (StateNow != PreviousState) {
  if (StateNow == LOW &amp;&amp; PreviousState == HIGH) //button pushed
  {
    digitalWrite(led1, LOW);
    digitalWrite(led2, LOW);
    digitalWrite(led3, LOW);
    digitalWrite(led4, LOW);
    Serial.println("Turned off all LEDs");
  }
  PreviousState = StateNow;
}


I simplify your code and add some comments:

#define led1 10 //10th,  
#define led2 11 //11th
#define led3 12 //12th
#define led4 13 //13th
#define ButtonA 9

int state;
int StateNow;
int PreviousState;

void setup() {
  pinMode(led1, OUTPUT);
  pinMode(led2, OUTPUT);
  pinMode(led3, OUTPUT);
  pinMode(led4, OUTPUT);
  pinMode(ButtonA, INPUT_PULLUP);
  Serial.begin(9600);
  Serial.println("Enter ch a, b, c, or d:");
  PreviousState = digitalRead(ButtonA);
}

void loop() {
  //detect changes in button signal to turn off leds
  StateNow = digitalRead(ButtonA);
  if (StateNow != PreviousState) {
    if (StateNow == LOW &amp;&amp; PreviousState == HIGH) //button pushed
    {
      digitalWrite(led1, LOW);
      digitalWrite(led2, LOW);
      digitalWrite(led3, LOW);
      digitalWrite(led4, LOW);
      Serial.println("Turned off all LEDs");
    }
    PreviousState = StateNow;
  }

  //check serial biffer to turn on leds
  if (Serial.available())
  {
    state = Serial.read();
    if (state == 'a' || state == 'A')
    {
      digitalWrite(led1, HIGH);
      Serial.println("Turned on led 1");
    }
    else if (state == 'b' || state == 'B')
    {
      digitalWrite(led2, HIGH);
      Serial.println("Turned on led 2");
    }
    else if (state == 'c' || state == 'C')
    {
      digitalWrite(led3, HIGH);
      Serial.println("Turned on led 3");
    }
    else if (state == 'd' || state == 'D')
    {
      digitalWrite(led4, HIGH);
      Serial.println("Turned on led 4");
    }
  }
}