Sections and categories in Joomla! provide an optional method for organizing your articles. Here's how it works. A Section contains one or more categories, and each Category can have articles assigned to it. One Article can only be in one Category and Section.
For example, you might have a Section called "Pets", and categories called "Dogs" and "Cats". Articles about dogs would be assigned to the "Dogs" Category, ones about cats the "Cats" Category. So an Article about dogs would be in the "Pets" Section and the "Dogs" Category. You could not have one Article that is in both the "Cats" and "Dogs" categories. To work around this, you could either (1) create a new Category called "Cats and Dogs" or (2) create a second Article to put in the second Category.
Why Use Sections & Categories?
There are two main reasons you might want to organize your Articles in sections and categories.
List and Blog Layouts
First, there are built-in Menu Item Types in Joomla! that take advantage of this organization. These are the Section Blog, Section List, Category Blog, and Category List. These Menu Item Types (also called "layouts") make it very easy to show articles that belong to sections or categories. As new articles are created and assigned to sections and categories, they will automatically be placed on these pages according to the parameters you set for each page.
For example, say you have a Section Blog layout for the "Pets" section, and say you have it set to order articles starting with the most recent one first. When you add a new Article to the "Pets" Section, it will automatically show on the "Pets" blog page as the first Article. You don't have to do anything other than add the Article and assign it to the "Pets" Section.
Organizing Articles in Article Manager
If you will have a large number of articles on your site, a second reason to use sections and categories is to simply group the articles so you can find them. For example, in the Article Manager, you can filter articles based on Section or Category. So if you have 200 articles in your site, you can more easily find an Article if you know it's Section or Category.
Sections and Categories vs. Menu Organization
It is important to understand that, just because Joomla! uses a 3-tier organization level for articles (Section → Category → Article), this has nothing to do with the structure of the menus on your site. For example, your site could have one menu level or six menu levels.
Other Types of Categories
There is also a potential for confusion about categories. Sections are only used for articles. However, categories are also used for other components, including Banners, Contacts, News Feeds, and Weblinks. These categories are completely different from Article categories and are set up in different screens in the back end of Joomla!. So when you see something about categories, it might refer to Article categories or it might refer to categories for these other components.
Creating a Section & Category Hierarchy