Let's start with some definitions. An Article is some written information that you want to display on your site. It normally contains some text and can contain pictures and other types of content. For many Joomla! sites, articles form the majority of the information presented in the website.
It is important to understand that, with Joomla!, the content of the site (for example, Articles) is totally separate from the formatting of the site -- the way it looks on the page. So it is best to think of articles as pure content, independent of the way it might be presented. For example, the same Article might be shown with different fonts, colors, headings, and background, and might be shown in different locations on the page.
Categories in Joomla! provide an optional method for organizing your articles. Here's how it works. A Category contains articles and other Categories. One Article can only be in one Category. If a category is in another category, it's called a subcategory of that category.
For example, you might have a Category called "Pets" and in that category, subcategories called "Dogs" and "Cats". Articles about dogs would be assigned to the "Dogs" Category, ones about cats to the "Cats" Category. You cannot 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 Categories?
There are two main reasons you might want to organize your Articles in categories.
List and Blog Layouts
First, there are built-in Menu Item Types in Joomla! that take advantage of this organization: Category Blog and Category List. These Menu Item Types (also called "layouts") make it very easy to show articles that belong to certain categories. As new articles are created and assigned to categories, they will be automatically placed on these pages according to the parameters you set for each page.
For example, say you have a Category Blog layout for the "Pets" Category, 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" Category, 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" Category.
Organizing Articles in Article Manager
If you will have a large number of articles on your site, a second reason to use categories is to simply group the articles so you can find them. For example, in the Article Manager, you can filter articles based on Category. So if you have 200 articles in your site, you can find an Article more easily if you know its Category.
Categories vs. Menu Organization
It is important to understand that the structure of Categories (e.g. Organisms -> Pets -> Animals -> Dogs) 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. Categories are also used by 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.
A Menu is a set of menu items like you see on the left hand side of this page. Each menu item defines an link to a page on your site which can be a simple article, a category list or an article blog. A Menulink has settings that control the contents and style of that page. Additionally, each menu item has an "alias" that will be used to create human-readable URLs when "Search Engine Friendly" URLs are on. Menu items can have any number of subitems.
A Menu isn't automatically shown on any page. You need to create the menu item to provide a link for your visitor to access your content.