These 16 recommended Drupal modules are not all recommended for every project, but they among the more useful modules out there. Many of them are useful only for site developers.
Admin Menu - An improved admin toolbar with drop down menus.
Backup Migrate - A must have for importing/exporting Drupal databases. Conveniently omits the data from the cache tables. Admins can schedule regular backups on a timed basis.
SEO Checklist - Helpful tool for identifying ways to improve site search engine efficiency.
Drush - Command line integration with your application. Modfiy variables, interact, and backup your site through a console.
Chaos Tools - A suite of tools for developers, including AJAX responders. This is a dependency for many modules.
Features - Package any content types, views, settings, and more into a downloadable feature. Features are modules that store components of the site. They are useful for re-using parts of a site for another, or for pushing up structural changes from a dev site to production site.
Libraries - Libraries API allows easy integration of 3rd party libraries through /sites/all/libraries. This is a dependency for many modules including colorbox and WYSIWYG.
Rules - Specify rules for things like when to send email notifications, custom redirects, system messages.
Token - Use token placeholders to output variables in content
Views - A must have for almost any site. Create and manage display views such as simple unordered lists, grids of images with thumbnails, javascript sliders, and much more.
WYSIWYG - "What You See is What You Get" framework module. Allows the integration of other WYSIWYG editors
Context - Create contexts for your site like 'front page', 'teens section', 'product detail page', in order to contextually control blocks or content output.
Entity Reference - Create a field for content types that allows reference to other content types. Very useful in many situtations when you want to attach one content type to another.
Calendar - Create nicely formatted calendars with events.
Date - Create date fields for content types.
Colorbox - Adding lightboxes to your site