Help, activity closes when launched

You must always add the Logcat to define the problem, since there may be several causes for your application to close.

I notice that your code is correct, the first thing is to ensure your Activity GreetingActivity, this registered in your AndroidManifest.xml

<activity android:name="SaludoActivity" />

Although the code is correct when reading the bundle in the second activity another detail that could close your application is that the value of "NAME" could be null, I recommend adding this validation:

 //Recuperamos la información pasada en el intent
    Bundle bundle = this.getIntent().getExtras();

if(bundle.getString("NOMBRE") != null){
    //Construimos el mensaje a mostrar
    txtSaludo.setText("Hola " + bundle.getString("NOMBRE"));
}

I see you're using Bundle and this is for fragments, with the intent is enough: use this

Intent intent = new Intent(MainActivity.this, SaludoActivity.class);
//Creamos la información a pasar entre actividades
intent.putExtra("NOMBRE", txtNombre.getText().toString());
//Iniciamos la nueva actividad
startActivity(intent);

Actity greeting (2nd activity)

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_saludo);
    final int nombre = getIntent().getIntExtra("NOMBRE", 0);
        Intent intent = getIntent();
        String nombreExtra = "";
      // Validar que este llegando 
        if (intent != null && intent.hasExtra("NOMBRE")) {
       //  get del parametro 
            nombreExtra = intent.getStringExtra("NOMBRE");
        }

    //Construimos el mensaje a mostrar
    txtSaludo.setText("Hola " + nombreExtra);
}
}

PS: we can improve your code based on this: 1. as NAME is a costante creates a static and immutable contact for this, 2. Also to respect the principle of Single Resposability let the put Extra within activity private static final String EXTRA_NAME="NAME";

Then your two Activity will be left like this:

public class SaludoActivity extends AppCompatActivity {

private static final String EXTRA_NAME = "NOMBRE";
private TextView txtSaludo;

public static Intent getStartSaludoActivityIntent(@NonNull Activity activity, String nombre) {
        Intent intent = new Intent(activity, SaludoActivity.class);
        intent.putExtra(EXTRA_NAME, nombre);
        return intent;
    }


@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_saludo);
    //Localizar los controles
    txtSaludo = (TextView)findViewById(R.id.txtSaludo);

final int nombre = getIntent().getIntExtra("NOMBRE", 0);
            Intent intent = getIntent();
            String nombreExtra = "";
          // Validar que este llegando 
            if (intent != null && intent.hasExtra(EXTRA_NAME)) {
           //  get del parametro 
                nombreExtra = intent.getStringExtra(EXTRA_NAME);
            }

        //Construimos el mensaje a mostrar
        txtSaludo.setText("Hola " + nombreExtra);   
}
}

and from your Main activity you call the static method

public class MainActivity extends AppCompatActivity {

private EditText txtNombre;
private Button btnAceptar;

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_main);

    //Obtenemos una referencia a los controles de la interfaz
    txtNombre = (EditText)findViewById(R.id.txtNombre);
    btnAceptar = (Button)findViewById(R.id.btnAceptar);

    //Implementamos el evento click del botón
    btnAceptar.setOnClickListener(new View.OnClickListener() {
        @Override
        public void onClick(View v) {
            //Iniciamos la nueva actividad
        startActivity(SaludoActivity.getStartSaludoActivityIntent(MainActivity.this, txtNombre.getText().toString()));


     }
        });
    }
}