In this blog we use Toolbar, it is more generalization form of ActionBar. It support more feature than ActionBar.
Toolbar is added in API level 21 but the good thing is we can use it in the below version by using supporting library.
First you to select the Theme.AppCompat.NoActionBar in styles.xml :-

<!-- Base application theme. -->
    <style name="AppTheme" parent="Theme.AppCompat.NoActionBar">
        <!-- Customize your theme here. -->
    </style>

Create a xml file with name toolbar.xml

<?xml version="1.0" encoding="utf-8"?>
<android.support.v7.widget.Toolbar
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    android:elevation="4dp"
    android:id="@+id/toolbar"
    android:background="#4082EE">

</android.support.v7.widget.Toolbar>

Create a xml file with name search_bar.xml in which we add EditText for search

<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:orientation="horizontal">

    <EditText
        android:id="@+id/edtSearch"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:imeOptions="actionSearch"
        android:inputType="text" />

</LinearLayout>

Now in the main layout (activity_main.xml) include toolbar.xml

<RelativeLayout 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=".MainActivity">

    <include
        android:id="@+id/toolbar"
        layout="@layout/toolbar"></include>

</RelativeLayout>

create menu_main.xml file inside menu folder

<item
    android:id="@+id/action_settings"
    android:orderInCategory="100"
    android:title="@string/action_settings"
    app:showAsAction="never" />

<item
    android:id="@+id/action_search"
    android:title="Search"
    android:orderInCategory="200"
    android:icon="@drawable/ic_menu_search"
    app:showAsAction="ifRoom"
    />

Now create your Activity like this:-

**
 * Activity that contains a custom Action bar with Material Design and search field
 */
public class MainActivity extends AppCompatActivity {


    private Toolbar toolbar;
    private MenuItem menuItem;
    private boolean isSearchOpened = false;
    private EditText searchEditText;

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

        toolbar = (Toolbar) findViewById(R.id.toolbar);
        setSupportActionBar(toolbar);
    }


    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        // Inflate the menu; this adds items to the action bar if it is present.
        getMenuInflater().inflate(R.menu.menu_main, menu);
        return true;
    }

    @Override
    public boolean onOptionsItemSelected(MenuItem item) {
        // Handle action bar item clicks here. The action bar will
        // automatically handle clicks on the Home/Up button, so long
        // as you specify a parent activity in AndroidManifest.xml.
        int id = item.getItemId();



        switch (id) {
            case R.id.action_settings:
                return true;
            case R.id.action_search:
                handleMenuSearch();
                return true;
        }

        return super.onOptionsItemSelected(item);
    }

    @Override
    public boolean onPrepareOptionsMenu(Menu menu) {
        menuItem = menu.findItem(R.id.action_search);
        return super.onPrepareOptionsMenu(menu);
    }

    /**
     * methods hides the keyboard
     *
     * @param view
     */
    private void hideKeyboard(View view) {
        //hides the keyboard
        InputMethodManager imm = (InputMethodManager) getSystemService(Context.INPUT_METHOD_SERVICE);
        imm.hideSoftInputFromWindow(view.getWindowToken(), 0);
    }

    /**
     * methods show the keyboard
     *
     * @param view
     */
    private void showKeyboard(View view) {
        InputMethodManager imm = (InputMethodManager) getSystemService(Context.INPUT_METHOD_SERVICE);
        imm.showSoftInput(view, InputMethodManager.SHOW_IMPLICIT);
    }

    /**
     * method handles the search logic
     */
    private void handleMenuSearch() {
        ActionBar action = getSupportActionBar(); //get the actionbar

        if (isSearchOpened) { //test if the search is open

            action.setDisplayShowCustomEnabled(false); //disable a custom view inside the actionbar
            action.setDisplayShowTitleEnabled(true); //show the title in the action bar

            hideKeyboard(searchEditText);


            //add the search icon in the action bar
            menuItem.setIcon(getResources().getDrawable(android.R.drawable.ic_menu_search));

            isSearchOpened = false;
        } else { //open the search entry

            action.setDisplayShowCustomEnabled(true); //enable it to display a
            // custom view in the action bar.
            action.setCustomView(R.layout.search_bar);//add the custom view
            action.setDisplayShowTitleEnabled(false); //hide the title

            searchEditText = (EditText) action.getCustomView().findViewById(R.id.edtSearch); //the text editor

            //this is a listener to do a search when the user clicks on search button
            searchEditText.setOnEditorActionListener(new TextView.OnEditorActionListener() {
                @Override
                public boolean onEditorAction(TextView v, int actionId, KeyEvent event) {
                    if (actionId == EditorInfo.IME_ACTION_SEARCH) {
                        doSearch();
                        return true;    
                    }
                    return false;
                }
            });

            searchEditText.requestFocus();

            //open the keyboard focused in the edtSearch
            showKeyboard(searchEditText);

            //add the close icon
            menuItem.setIcon(getResources().getDrawable(android.R.drawable.ic_menu_delete));

            isSearchOpened = true;
        }
    }

    @Override
    public void onBackPressed() {
        if(isSearchOpened) {
            handleMenuSearch();
            return;
        }
        super.onBackPressed();
    }

    private void doSearch() {
//
    }
}