Categories are the way you organized your articles in Joomla. In past versions there was a concept of "Sections" and "Categories" where a category was a child of a section. This was basically a two-level structure which definately had its limitations. Starting in Joomla 1.6, you only have categories. But the cool thing is that you can make any category a child or parent of any other category, and from what we've seen there really is no limit to the number of children a category can have.
So needless to say, you should put a little thought into how you would like to structure your categories before you go just randomly creating them. This could save you a pretty big headache down the road as your site grows and you add more and more content. That said, caution should be used to not over think this as well. Creating a structure that is so complex you can't even remember it can be bad too. A little common sense is all you really need.
To edit an article you can either:
By default there is one category already created for you. Its the "uncategorized" category. This is where all articles will go until you create another category.
You will notice this looks similar to the Article Manager discussed earlier. Many of the same features exist here, so we won't go into that detail again.
The important thing here is that you can "Create a new Category" by clicking the "New" button at the top of the page in the Control Toolbar
The Add Category screen should also look similar to the Add Article screen
Your options here are to:
Once you have made all your initial changes, click either "Save", "Save & Close", "Save & New" or "Close". This functionality is the same as it is for articles.
The Edit Category screen is exactly the same as the Add Category screen, with the exception that the existing values are already populated. Change them if you like, and save your changed.