This tutorial explain how we can Play YouTube video within Fragment.
 

1. Create simple Fragment class extending Fragment.

2. Implement YouTubePlayer.OnInitializedListener to Fragment and override its methods.

3. Create FrameLayout in Layout file of Fragment. like below:-
 

<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    tools:context="com.ammy.youtubesample.MainFragment">

    <FrameLayout
        android:id="@+id/frame_fragment"
        android:layout_width="match_parent"
        android:layout_height="200dp"></FrameLayout>

</FrameLayout>

4. In Fragment Java class Create YouTubePlayerFragment instance and replace this fragment with framelayout like below:
 

 @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container,
                             Bundle savedInstanceState) {
        View view =  inflater.inflate(R.layout.fragment_main, container, false);

        YouTubePlayerFragment youtubePlayerFragment = new YouTubePlayerFragment();
        youtubePlayerFragment.initialize(getString(R.string.api_key), this);
        FragmentManager fragmentManager = getFragmentManager();
        FragmentTransaction fragmentTransaction = fragmentManager.beginTransaction();
        fragmentTransaction.replace(R.id.frame_fragment, youtubePlayerFragment);
        fragmentTransaction.commit();


        return view;
    }

5.  Check initialization of youTube in listener's method and play video.
 

 @Override
    public void onInitializationSuccess(YouTubePlayer.Provider provider, YouTubePlayer youTubePlayer, boolean b) {
        if(!b){
            youTubePlayer.cueVideo("EJylz_9KYf8");
        }
    }

    @Override
    public void onInitializationFailure(YouTubePlayer.Provider provider, YouTubeInitializationResult youTubeInitializationResult) {
        Toast.makeText(getActivity(),"fail",Toast.LENGTH_SHORT).show();

    }

Enjoy.
Happy Coding... ;)