Categories in Joomla provide a method for organizing your content/articles. Here's how it works. Each Category has Articles assigned to it, and some categories may have sub-categories (think parent/child organizational structure). 

Another way to consider this organization is to have a Category called “Pets,” and two sub-categories, one called “Dogs” and one called “Cats.”  Articles about choosing a pet would go under the general category of "Pets". Articles about dogs would be assigned to the “Dogs” sub-category and ones about cats the “Cats” sub-category. You could not have one article that is in both the “Cats” and “Dogs” categories.

Why Use Categories?
There are two main reasons you might want to organize your articles in sections and categories. These are listed here, but will be described in detail in a later document.
  1. List and Blog Layouts
  2. Organizing Articles in Article Manager
Create a Category:
Navigate to the Content menu. In the drop-down list a menu item named Category Manager will appear.  Click it.
How to navigate to the category manager
Once you reach the Category Manager, on the top right you will see several icons.

Click the one marked New to create a new Category.

Locate New icon
On the next screen you need to type in a title, and then optionally an alias. 
SUGGESTION: Make the Category title Category 5.
TIP: If you do not add an alias, whatever you put in as the title will become the alias, including unimportant words such as ‘a,’ ‘and,’ and ‘the.’   Failure to add an alias will result in an alias being created for you based upon your category title.

Once you have a title and alias, click "Save & Close."   For example:
Enter category info
After you’ve saved the new category you will be taken to the Category Manager. You can create as many nested categories (or subcategories) as you want.  Let's add one subcategory to our above example.  Within the Category Manager, click on New.  Create a title and and alias.