Let us say you have thought of a new category you want to place some articles in. To create this new category, go to one of the pages that you wish to put there, and add a category tag naming the new category to the end of the article, like this:
[[Category:Category name]]
...where in place of category name you type the actual name of the category. When you save the page, the category should appear on the bottom line of the page. If it is indeed a new category, it will turn up in red. But this does not mean there is no such category: it might exist but with a slight difference in naming. Before you create a new category, make sure it does not already exist. In a new browser window, click on Special pages in the toolbox menu on the left side of your screen. Then click on All pages. Pick Category from the Namespace dropdown menu, and then enter the name of the category. Look over the index for synonymous categories.
Once you are sure your new name is good to go: Click on the redlink, and then click on the article creation link provided in the instructions that appear on your screen.
You will need to put a parent category on the new page, and then save the page. Going forward you can put your newly named category at the bottom of pages that you wish to add your new category to. Categories with too few pages in them are usually nominated for deletion.