How to implement a Subscribe block functionality? I wonder how to implement a block similar to the block shown below, to integrate with the Subscriptions or Notification module.

Basically it sends emails to user everytime a new article is added.



Any suggestions?
Assuming this question is about D7 (at least), I wouldn't recommend using either of both modules anymore. This because these days there is a way better alternative available, which is the Message Stack. It consists of these modules (quotes are from the module's project page):

Message:

The Message module is the core of the message stack. It enables logging and displaying system events in a number of different use cases. Events that are recorded over time are sometimes call activity streams. Exportable messages subtypes can be created for different use cases with custom fields and display (view) modes.


Message Notify.

This module provides a method for sending a message via a notifier plugin. Message Notify comes with plugins for email and SMS and may be extended to other transport mechanisms as required.


Message Subscribe.

With this module, users who subscribe to content will be notified when events occur that involve that content. The module leverages the Flag module, which provides a subscribe functionality to users.



These are the basic components and dependencies of the message stack:

Quite a few popular Drupal distributions already adopted to this Message Stack, such as such as Commerce Kickstart, Drupal Commons, ERPAL and Open Atrium. And it will make you well positioned for Drupal 8 also (because of the modules' object orientated architecture).
If you want to have a closer look at the Message Stack in your own environment, then experiment a bit with the Answers module (disclosure: I'm a co-maintainer), together with the Answers Notification sub-module, which allows logged in users to subscribe to selected questions, so that they receive notifications (using the Message Stack) when Questions receive Answers.
Attention: the Message Stack is really amazing (I think), but also not obvious to get started with it. Especially if you try to do so only using site building techniques. But after you climbed this Drupal Message ladder, you might not ever want to consider building a site without it ... trust me.