This web part shows the current logged in user what lists or libraries the user had subscribed for it.
It will show gold bell icon beside the list name; which means you subscribed for this list.
If there is silver colour bell beside the list name; this means you didn’t subscribe to this list. To subscribe, you can click the list name, it will popup the “New Alert” SharePoint OOB model dialog.
Also in this dialog you will have many options of when to receive alerts and on what changes exactly.
In the Tool Part of the Web Part, user can select the lists on current site that the user have permission to only to see it displayed in the web part to subscribe to it.
Below is screen shot:
Contact me at email@example.com for more information on this and other custom developed SharePoint Web Parts and Apps