You can extend the core functionalities provided by Drupal by installing modules. This is just some of the common features and is neither a complete list nor a significant portion of the complete list. You can also find the module or modules which can be used to attain a particular feature here for the features listed here. This is just to show what is possible; we will have separate lists for modules separated into categories along with their usage details.
Integrate with facebook through Like Button, Comments, Like box, Recommendations, Facepile, Live stream, Activity Feed, Send button, Subscribe button etc.
Module: FB social
Module page: https://drupal.org/project/fb_social
Add the Google +1 (Plus One) out-of-the-box to your nodes, or as a block without the need to copy/paste anything.
Module: google plus one
Module page: https://drupal.org/project/google_plusone
An easy way for users to bookmark and share your site and/or pages with various internet service providers such as Facebook, Gmail, google+, twitter, delicious, Google, Orkut etc. using a single click interface.
Module: add this
Module page: https://drupal.org/project/addthis
Use a WYSIWYG (What You See Is What You Get) for easy formatting of the content.
Module page: https://drupal.org/project/ckeditor
Automatically generate URL alias for contents so that users don’t have to manually specify URL alias.
Module page: https://drupal.org/project/pathauto
Send notification e-mails to visitors about new, published comments on pages where they have commented.
Module: Comment notify
Module page: https://drupal.org/project/comment_notify
Backup and restore MySQL database and site files, either manually or through scheduled automatic backups.
Module page: https://drupal.org/project/backup_migrate
Create a TODO list for better SEO (Search Engine Optimization) and track the progress of the TODO list.
Module page: https://drupal.org/project/seo_checklist
Provide an easily accessible theme-independent administration interface over pages with regular menu items, tasks, and actions, to enable fast access to any administrative resource your Drupal site provides.
Module: admin menu
Module page: https://drupal.org/project/admin_menu
Implement a module filter capability to quickly find the modules without having to rely on the browsers search feature, especially when the number of modules is more.
Module: module filter
Module page: https://drupal.org/project/module_filter
Create frequently answered question-answer pairs which can be easily displayed on the 'faq' page.
Module page: https://drupal.org/project/faq
Create a more professional discussion forum similar to the ones available in most websites extending the core forum functionality provided by Drupal.
Module: advanced forum
Module page: https://drupal.org/project/advanced_forum
Differentiate between internal and external links and give warnings and/or open in a new tab automatically when an external link is clicked.
Module: external links
Module page: https://drupal.org/project/extlink
Implement a regular CAPTCHA, which is a challenge-response test placed within web forms to determine whether the user is human.
Module page: https://drupal.org/project/captcha
Create an advanced Captcha that lets site administrators create their own questions to foil automated spam bots (e.g. what is 5+2 or what is the name of this site or who is the owner of this site etc.).
Module: captcha riddler
Module page: https://drupal.org/project/riddler
Add arbitrary moderation states and node revisions, and moderate their transitions through a workflow approval process.
Module: Workbench moderation
Module page: https://drupal.org/project/workbench_moderation
Create editorial access controls based on hierarchies which can be used for news portal websites.
Convert an existing content type to another content type after the creation of the content.
Module: Node convert
Module page: https://drupal.org/project/node_convert
Create advanced user profiles similar to most professional user-driven websites extending the core functionality provided by Drupal.
Module page: https://drupal.org/project/profile2
Implement an advanced search that can query the core and extended profile fields.
Implement a configurable mailing list system where mailing lists can be created and add users along with their signup or through a batch processing.
Create a community portal where users can create and manage their own 'groups'. Each group should have subscribers and maintain a group home page where subscribers can communicate amongst themselves.
Module: Organic groups
Module page: https://drupal.org/project/og
Create a rewards point (user points) system through which users can gain or lose points for performing certain actions on your site.
Module: User Points
Module page: https://drupal.org/project/userpoints
Automatically detect the type of device (desktops, mobile phones, tablets etc.) and render different looks (theme) for different devices.
Module: Mobile Theme
Module page: https://drupal.org/project/mobile_theme
Create a quick finder menu with each menu item being a tag. Content should be allowed to be tagged with multiple topics (tags) and each topic (tag) should become a menu item in the quick finder menu. Once you click a menu item (topic tag) on the quick finder menu, all the contents tagged with that topic should be displayed. Alter the default display view of blocks and contents and add functionalities like sort blocks differently, change the way a particular content type display data, display a block with the N most recent posts of some particular type etc.
Module: Taxonomy Menu
Module page: https://drupal.org/project/taxonomy_menu