One of the best ways to get to know Drupal is to find the useful parts and, well, use them! Here are the most popular modules in Drupal 6, according to the stats at Drupal.org.
10. Google Analytics
The easiest way to add Google Analytics to your site is probably to paste some HTML into your Drupal template file, but by doing this you miss out on the powerful features of Drupal's Google Analytics module, including role-based tracking and audience segmentation. You can cache the Google Analytics files locally to help your page load times, which can help your Yslow rating. I like to exclude admin users from tracking, so that I'm not influencing my page counts while I navigate around my own site.
The IMCE module lets admins and users upload and browse files. It integrates with a bunch of graphical editors through the IMCE-WYSIWYG API bridge module (what a mouthful!) Note: If you use IMCE with Google Analytics or Securepages, make sure to add the IMCE module directory to your list of exceptions. These modules don't play nice together!
Hey, it may not be sexy, but it gets the job done. The date project is a collection of modules that store, well, date information. Timestamps, timezones, UTC, GMT, CCK integration, calendar browsers... this is a pretty compelling and comprehensive package.
7. ImageAPI and ImageCache
ImageAPI and ImageCache are very popular building blocks that other modules depend upon. They are both so useful, in fact, that most of the functionality they provide will be built into Drupal 7 core.
6. Admin Menu
Navigating through Drupal's complex and hierarchical administration pages can be a nightmare, especially if you are unlucky enough to have a theme that doesn't use breadcrumbs. Enter the Administration Menu module, which provides a cross-platform theme-independent dropdown menu at the top of your browser, allowing you to deep-link to every admin page with one click. This module is an absolute necessity for me.
5. FileField and ImageField
These modules both provide CCK fields that you can attach to any custom content type. FileField provides a CCK-compatible alternative to the core Upload module, which helps you attach one or more files to a node. ImageField is closely related, and depends on FileField to allow users to upload images, which are then scaled using ImageCache (see above.) Great for providing clients with an easy, visual way to upload files and images. Both of these will be in Drupal 7 core.
Everybody's favorite Drupal SEO module, Pathauto helps improve your Drupal site's pagerank. Instead of automatically assigning your nodes URLs like node/123, or forcing you to type in an alias for each node you create, Pathauto creates an alias automatically based on factors such as the node title, taxonomy terms, or even authors. You choose!
Not sexy, but indispensable: this is the token module's mantra. Token provides the ability for other modules to quickly replace bits of text with other bits of text, which might not seem that impressive until you see the list of modules that use Token. If you're interested in the abstract concept of a token, Wikipedia has all the gory details. Token will be included in Drupal 7.
CCK is an extremely powerful and robust way for developers and administrators to create their own content types without touching a line of code. It's used by a number of very, very popular modules and part of it will be in Drupal 7 core (although the module will still exist in Drupal 7.) You can still create your own custom content types by coding them up, but for most applications, using CCK is faster and easier.
And now, drumroll please, the most popular module in Drupal 6 is...
Views has become the real power behind Drupal. You can use them to display lists of content, tables of content and even maps of content, and have those lists be sortable, filtered according to different arguments, or transformed in almost any way you can imagine. Views is an invaluable resource for deciding which data to present and how to present it, and it mines Drupal's database without requiring you to write any SQL or even any code at all!