The most simplistic approach

Let's begin by illustrating the most common approach newcomers take in order to solve this problem. It starts by enumerating all the possible states a game could have:

enum class StateType{
    Intro = 1, MainMenu, Game, Paused, GameOver, Credits
};

Good start. Now let's put it to work by simply using a switch statement:

void Game::Update(){
  switch(m_state){
    case(StateType::Intro):
      UpdateIntro();
      break;
    case(StateType::Game):
      UpdateGame();
      break;
    case(StateType::MainMenu):
      UpdateMenu();
      break;
    ...
  }
}

The same goes for drawing it on screen:

void Game::Render(){
  switch(m_state){
    case(StateType::Intro):
      DrawIntro();
      break;
    case(StateType::Game):
      DrawGame();
      break;
    case(StateType::MainMenu):
      DrawMenu();
      break;
    ...
  }
}

While this approach is okay for really small games, scalability here is completely out of the question. First of all, the switch statements are...