NotificationListenerService was introduced from Android team as part of API 18 (Android 4.3), this class allows an application to receive information from notifications when it's created and removed.
To use it we need to implement 2 callback methods. Both of them have paramether named sbn which is object of StatusBarNotification class.
In this example I will show quickly how to listen notification events with just print them to console.logs
Let's start with creating new Java Class named NotificationService.java with following content:

package me.yuks.demo;

import android.app.NotificationManager;
import android.content.Context;
import android.content.Intent;
import android.os.Bundle;
import android.service.notification.NotificationListenerService;
import android.service.notification.StatusBarNotification;
import android.util.Log;
import android.support.v4.content.LocalBroadcastManager;


public class NotificationService extends NotificationListenerService {

    Context context;

    @Override

    public void onCreate() {

        super.onCreate();
        context = getApplicationContext();

    }
    @Override

    public void onNotificationPosted(StatusBarNotification sbn) {


        String pack = sbn.getPackageName();

        String text = "";
        String title = "";
        if (android.os.Build.VERSION.SDK_INT >= android.os.Build.VERSION_CODES.KITKAT) {
            Bundle extras = extras = sbn.getNotification().extras;
            text = extras.getCharSequence("android.text").toString();
            title = extras.getString("android.title");
        }

        Log.i("Package",pack);
        Log.i("Title",title);
        Log.i("Text",text);

    }

    @Override
    public void onNotificationRemoved(StatusBarNotification sbn) {
        Log.i("Msg","Notification was removed");
    }
}

Also we need to update our Manifest file to include this newly created service.

<service
    android:name=".NotificationService"
    android:enabled="true"
    android:icon="@mipmap/ic_launcher"
    android:label="@string/app_name"
    android:permission="android.permission.BIND_NOTIFICATION_LISTENER_SERVICE">
    <intent-filter>
        <action android:name="android.service.notification.NotificationListenerService" />
    </intent-filter>
</service>

So basically that is :) Run and enjoy :) Note: User require to enable notification permission from "Settings > Security > Notification access".