Navigation Drawer Below Toolbar

You should move DrawerLayout as top parent and move Toolbar out of DrawerLayout content container.
In short this looks like:

RelativeLayout
 ----Toolbar
 ----DrawerLayout
     ---ContentView
     ---DrawerList

<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:id="@+id/top_parent"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:fitsSystemWindows="true"
    tools:context=".MainActivity">

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

    <android.support.v4.widget.DrawerLayout
        android:id="@+id/drawer_layout"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:layout_below="@+id/toolbar">

        <FrameLayout
            android:id="@+id/content_frame"
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            android:background="@color/background_color" />

        <ListView
            android:id="@+id/drawer"
            android:layout_width="260dp"
            android:layout_height="match_parent"
            android:layout_below="@+id/toolbar"
            android:layout_gravity="start"
            android:layout_marginTop="56dp" />

    </android.support.v4.widget.DrawerLayout>
</RelativeLayout>

However, Material Design guidelines state that Navigation Drawer should be above the Toolbar.

How to put navigation drawer below toolbar?

certainly android:layout_marginTop="?attr/actionBarSize" do the job in

<android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/my_drawer_layout"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:layout_marginTop="?attr/actionBarSize">

But the problem is drawerlayout is top of the toolbar. That is why the fading here.
you can remove fading by

mDrawerLayout.setScrimColor(getResources().getColor(android.R.color.transparent));

But on some devices it may look wired.

Solution

When working with Android studio. We can create NavigationDrawerActiviity
There are 3 files named

activity_main.xml

app_bar_main.xml

nav_header_main.xml

content_main.xml

So we can skip app_bar_main.xml and we can remove the fading.

Step 1

Make the root view of the activity main as Vertical LinearLayout

<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical"
android:fitsSystemWindows="true"
tools:context="com.example.MainActivity">
</LinearLayout>

In activity_main.xml add DrawerLayout and include content_main.xml in DrawerLayout. and Add AppBarLayout above the DrawerLayout.

<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical"
tools:context="com.qproinnovations.schoolmanagement.activity.HomeActivity">
    <android.support.design.widget.AppBarLayout
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:theme="@style/AppTheme.AppBarOverlay">

        <android.support.v7.widget.Toolbar
            android:id="@+id/toolbar"
            android:layout_width="match_parent"
            android:layout_height="?attr/actionBarSize"
            android:background="?attr/colorPrimary"
            app:popupTheme="@style/AppTheme.PopupOverlay" >

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

    </android.support.design.widget.AppBarLayout>
    <android.support.v4.widget.DrawerLayout
        android:id="@+id/drawer_layout"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:fitsSystemWindows="true"
        tools:openDrawer="start">
    <!-- drawer view -->
        <include layout="@layout/content_main" />
<!-- drawer content -->
        <android.support.design.widget.NavigationView
            android:id="@+id/nav_view"
            android:layout_width="wrap_content"
            android:layout_height="match_parent"
            android:layout_gravity="start"
            android:fitsSystemWindows="true"
            app:menu="@menu/activity_home_drawer" />
    </android.support.v4.widget.DrawerLayout>

</LinearLayout>

Step 2

add and replace setContentView() of NavigationDrawerActiviity to

setContentView(R.layout.activity_main);

Finally we have

enter image description here

Avoid Toolbar overlain by NavigationDrawer

Roughly saying, you can. Wrap your layout/activity_main.xml with LinearLayout, and move AppBarLayout to there from layout/app_bar_main.xml.

layout/activity_main.xml:

<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    xmlns:tools="http://schemas.android.com/tools"
    android:fitsSystemWindows="true"
    android:orientation="vertical"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

    <com.google.android.material.appbar.AppBarLayout
        android:layout_width="match_parent"
        android:layout_height="wrap_content"
        android:theme="@style/AppTheme.AppBarOverlay">

        <androidx.appcompat.widget.Toolbar
            android:id="@+id/toolbar"
            android:layout_width="match_parent"
            android:layout_height="?attr/actionBarSize"
            android:background="?attr/colorPrimary"
            app:popupTheme="@style/AppTheme.PopupOverlay" />

    </com.google.android.material.appbar.AppBarLayout>

    <androidx.drawerlayout.widget.DrawerLayout
        android:id="@+id/drawer_layout"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        tools:openDrawer="start">

        <include
            layout="@layout/app_bar_main"
            android:layout_width="match_parent"
            android:layout_height="match_parent" />

        <com.google.android.material.navigation.NavigationView
            android:id="@+id/nav_view"
            android:layout_width="wrap_content"
            android:layout_height="match_parent"
            android:layout_gravity="start"
            app:headerLayout="@layout/nav_header_main"
            app:menu="@menu/activity_main_drawer" />

    </androidx.drawerlayout.widget.DrawerLayout>
</LinearLayout>

layout/app_bar_main.xml:

<?xml version="1.0" encoding="utf-8"?>
<androidx.coordinatorlayout.widget.CoordinatorLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"
    tools:context=".MainActivity">

    <include layout="@layout/content_main" />

    <com.google.android.material.floatingactionbutton.FloatingActionButton
        android:id="@+id/fab"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_gravity="bottom|end"
        android:layout_margin="@dimen/fab_margin"
        app:srcCompat="@android:drawable/ic_dialog_email" />

</androidx.coordinatorlayout.widget.CoordinatorLayout>

Though you may need a few more style adjustements for side effects (those may be posted as other new questions), this time your question itself:

I want not to overlay the toolbar.

should be fulfilled.

For system window's color adjustment:

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_main);
    getWindow().setStatusBarColor(ContextCompat.getColor(this, R.color.colorPrimaryDark));

    (...)
}

screenshot

Put Navigation Drawer below Toolbar and above TabLayout

Instead of using DrawerLayout as root Create RelativeLayout as root and put DrawerLayout below Toolbar inside RelativLayout something like:

<?xml version="1.0" encoding="utf-8"?>

<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:fitsSystemWindows="true"
    android:orientation="vertical">

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

    <android.support.v4.widget.DrawerLayout
        android:id="@+id/drawer_layout"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:layout_below="@+id/toolbar">

        <LinearLayout
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            android:orientation="vertical">

            <com.astuetz.PagerSlidingTabStrip
                android:id="@+id/vp_indicator"
                android:layout_width="match_parent"
                android:layout_height="48dp"
                android:background="@color/colorPrimary"
                app:pstsIndicatorColor="@color/white"
                app:pstsIndicatorHeight="2dp"
                app:pstsShouldExpand="true"
                app:pstsTabTextColor="@drawable/switcher_indicator_text_selector"
                app:pstsTabTextSize="@dimen/text_size_large"
                app:pstsUnderlineColor="@color/colorPrimary"
                app:pstsUnderlineHeight="2dp" />

            <android.support.v4.view.ViewPager
                android:id="@+id/vp_pager"
                android:layout_width="match_parent"
                android:layout_height="match_parent" />

        </LinearLayout>

        <include layout="@layout/include_navigation_view" />

    </android.support.v4.widget.DrawerLayout>

    <FrameLayout
        android:id="@+id/frame_fragment_container"
        android:layout_width="match_parent"
        android:layout_height="match_parent" />

</RelativeLayout>

How to put navigation-drawer under app-bar for mobile in vuetify?

This worked for me.

Vuetify navigation drawer under app-bar

<v-app id="inspire">
   <v-overlay
      :value="drawer"
      z-index="4"
    >
    </v-overlay>
    <v-navigation-drawer
      v-model="drawer"
      app
      clipped
      hide-overlay
      :style="{ top: $vuetify.application.top + 'px', zIndex: 4 }"
    >
    </v-navigation-drawer>

    <v-app-bar
      app
      clipped-left
      dark
      color="#447fa6"
    >
      <v-app-bar-nav-icon @click.stop="drawer = !drawer"></v-app-bar-nav-icon>
      <v-toolbar-title>Application</v-toolbar-title>
    </v-app-bar>
</v-app>

Codepen

How do I make DrawerLayout to display below the Toolbar?

<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
android:orientation="vertical"
android:layout_width="match_parent"
android:layout_height="match_parent">

    <!-- The toolbar -->
    <android.support.v7.widget.Toolbar  
        android:id="@+id/my_awesome_toolbar"
        android:layout_height="wrap_content"
        android:layout_width="match_parent"
        android:minHeight="?attr/actionBarSize"
        android:background="?attr/colorPrimary" />

    <android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/my_drawer_layout"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

        <!-- drawer view -->
        <LinearLayout
            android:layout_width="304dp"
            android:layout_height="match_parent"
            android:layout_gravity="left|start">

            <!-- drawer content -->

        </LinearLayout>

        <!-- normal content view -->
        <LinearLayout
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            android:orientation="vertical">

            <!-- The rest of content view -->

        </LinearLayout>  
    </android.support.v4.widget.DrawerLayout>

</LinearLayout>

Navigation Drawer Below Toolbar