Develop Reference

Is it mandatory to use GCM service to show push notification?

I am developing a Cordova app having a Java backend. I want to know if it's mandatory to use GCM service in order to show a push notification in Android/iOS (in notification tray)?
I am using Apache Kafka, MQTT brokers in my backend. Is it possible to show push notification using these solutions, i.e. using completely self defined solutions?
Using MQTT & other brokers I am able to send realtime data to users but how to show those message as native notification in notification bar is my problem.

Yes, you will need to use GCM to send push notifications to Android devices. You can even use GCM to send push notifications to iOS devices (GCM makes request to APNS (Apple push notification service) to make it work).
But GCM for iOS isnt trusted much.
You can use any backend to send push notification to your devices. You just need to make a request to GCM/APNS.
This is the cordova plugin for push notification.
Integrating with GCM is damn easy, but it takes a little effort with APNS.
Most people use third party paid services like Pushwoosh to implement their push backend because its convenient. There is a free alternative OneSignal but it tracks your usage.
Follow this guide to setup push for APNS. (You can just use the Setup certificate part from this guide, since you are not using C#)

To receive push you need use/implement GCM(and actually FCM, because of GCM deprecation)
To show notification from app it is unnecessary to implement GCM/FCM

Can I Use Application Web Services Which is Already Made for Android in iOS especially in push notification

Is their any difference between web services for android and iOS ?? especially if i am implementing push notification using google cloud messaging
Thank you in advance

Web services
There shouldn't be any major differences between web services for Android and iOS. Especially when it comes to exchanging JSON with object representations. At the end, it's all the same, parsing and mapping. Sometimes you might want to return some special, platform specific information. In this case, the mobile clients need a way to tell your backend which OS they're running.
Push notifications
Push notifications is a different story. On Android you have Google Cloud Messaging and on iOS Apple Push Notification Service. Your back end has to know which notification service to use.
There are many ways to solve this problem.
One way is to set a custom HTTP header that would contain information about the OS the mobile client runs. Another is to have a parameter that is sent with each request.

implementing GCM communication

I want to implement GCM communication in order to create push notifications, from my application server to all the devices that my app is installed via GOOGLE's servers.
I have found a library that helps me do this the Parse library and using the dashboard given I've sent a push notification from server to device and vice-versa.
The thing is that I want to be able to use the push notification (server side) from my website and not having to log in in Parse's. I'm pretty clueless about this, So I'd appreciate any direction and suggestion given

You may start with these tutorial
Using asp.net
Redth/PushSharp - A server-side library for sending Push Notifications to iOS (iPhone/iPad APNS), Android (C2DM and GCM - Google Cloud Message), Windows Phone, Windows 8, Amazon, Blackberry, and (soon) FirefoxOS devices!
Push Messages to GCM server using visual c#.net application (server side).
And using PHP and MySQL
Android Push Notifications using Google Cloud Messaging (GCM), PHP and MySQL
And using Parse API
if you want to send pushnotification from parse API (not dashboard) then see
Push Notification Guide

There are docs in Parse for sending push notification from Ios,Android,.net,Javascript,php and REST services, use one them which is appropriate for you.
https://parse.com/docs/push_guide#top/JavaScript

How do Asp.net backend and Web API web service hosted on Azure send push notification to mobile app?

I am building a mobile app, for both Android and iPhone. I outsourced the mobile end to an app company, while I myself develop the back end and web service part with asp.net and hosted them on Azure. The web service is developed with Web API 2.0, and it is deployed to Azure as a web site. The back-end is developed with asp.net web forms and deployed in to Azure as a web site.
There are 2 cases when a mobile user will receive some messages:
(1) When some mobile users post some messages, other mobile users will receive them.
(2) In the back end, if an administrator publishes a message, all mobile users will receive it.
Currently in my back end and web service, when mobile user post message or an administrator publishes a message, I just update database table to save that message. I thought the mobile end will keep pulling database table and get the latest message - until today. Today the app company told me I need to PUSH those messages to phones. He suggested some third party service to achieve this.
I then checked Azure documents, it seems they have built-in push function in their service so that I don't need go for a third party service. But I am confused, I saw there is "Cloud Services", there is "Mobile Services", and there is "Service Bus-Notification Hub". Which one should I use to make my back end and web service be able to send push notification to mobile devices? How should I modify my existing back end and web service projects in Visual Studio Express 2013? Do I still deploy them to Azure as web sites?
by message, I don't mean SMS, I mean some text data.

Given that you've already invested some reasonable development time in both the Android Application and your back-end services, it sounds like you're looking for Notification Hubs. Notification Hubs are designed specifically for sending push notifications to both individual users and groups of users across one or more mobile platforms.
As suggested in my comment, Cloud Services is a means of hosting your application on Azure (for your purposes it would offer a heavier-weight alternative to Azure Websites with an improved SLA), and Mobile Services provides a set of libraries and APIs to abstract interacting with your application's server-side data.
To use a Notification Hub to send push notifications you'll need to do a few things (described in detail in the linked documentation):
Go to the portal and create yourself a new Notification Hub
Create a Google Cloud Messaging Project via the Google Cloud Console (Azure will use this to transport your message to the relevant Android devices)
Update your back-end service to use Notification Hub to send messages
Modify your android application to use the the associated Azure messaging libraries to receive your push notifications
I hate to link directly to documentation, but the best way to get started with this functionality is via the following tutorial from the Azure documentation:
http://azure.microsoft.com/en-us/documentation/articles/notification-hubs-android-get-started/. This covers the basic changes you'll need to make to both your Android application to handle push notifications, and to your back-end service in order to send them.
You'll still be able to deploy your server-side components to Azure Websites.
Note that sending messages via Notification Hub comes with a price tag, though there is a free tier that limits you to 500 devices and 100,000 push notifications per month, as described in http://azure.microsoft.com/en-us/pricing/details/notification-hubs/.
EDIT: Here's an example of using Notification Hub to send messages to individual users:
http://azure.microsoft.com/en-us/documentation/articles/notification-hubs-aspnet-notify-users/

Right way to establish push notifications for Android/iOS

Assuming my application is based on HTML5/Jquery mobile (Cross platform web app),
How is it possible to create push notifications for Android/iOS ?
Also, does it include developing different "notifications engine" for
each device? (in other words: Do I need to implement two different background processes for each type of device?)
I'd be really glad to get a full answer/references to my conflict.
I have searched for the forum's previous answers but it seems like the threads are quite old so i preferred opening my own question.

GCM is the Google Cloud Messaging Service for Android, while APNS is the push notification service for iOS, both of them are very similar.
If you're already using html5 and jQuery, I would suggest you to use Phonegap which is a framework that allows you to create mobile apps using only html5 css3 javascript(jQuery) it was very useful for me and they've a push notifications plugin which works for Android and iOS.
Be sure to check Phonegap and Phonegap Build (which is a very interesting service)
Regards

1)
Apple has Apple Push Notification Service (APNS) to push out alerts.
More details at
https://developer.apple.com/library/ios/documentation/NetworkingInternet/Conceptual/RemoteNotificationsPG/Chapters/ApplePushService.html.
Google offers a comparable Google Cloud Messaging Service (GCM).
Details at http://developer.android.com/google/gcm/index.html. UPDATE: GCM does support iOS as a client, so can be used for both Android and iOS, with some work. (Technically, it will be an APNS when it reaches the iOS device.)
2) Not sure I understand what you mean by "background processes", If you are referring to backend service/originator of notifications,then the service needs to be able to communicate i with both the Apple APNS or Google CDM using the interfaces supported by these services and send out alerts in appropriate format. If you are looking for a third party service for sending out push notifications, you can checkout UrbanAirship or Parse which can send notifications to either platform.