A category is a software feature of MediaWiki. Categories provide automatic indexes that are useful as tables of contents.
To add a page to a category for example, type:
To be specific, in order to add an article called "Albert Einstein" to the category "People", you would edit the article and add [[Category:People]] into the page source somewhere. Exactly where doesn't matter, but the convention is to put it after the article text.
A category page does not need to be created beforehand. Putting a category link in one of the articles will automatically create that category and display the page (or pages) that belongs to it. By looking at the very bottom of the any article, you can tell what categories it belongs to, if any. Categories, if they exist, are always displayed there. Although a category has multiple articles in it, it will appear as a red link when a description of the category page is empty. To make the category links appear blue (like existing articles), click on the category link to go to the category page and edit it to add a short description.
Categories can have sub-categories, meaning that you can put a category page inside of another category. Categories can be nested in multiple level in this fashion. For example "Social Sciences" category may contain a list of categories such as "Sociology", "History", etc... Only after clicking on one of them, you would get the list of articles in that particular sub-category.
For a complete list of categories, see Special:Categories.
To link to a category page from the article text (without adding the page to the category) prefix the link name with a colon. For example, [[:Category:Out_and_Elected|Out and Elected category]] renders as Out and Elected category.
Categories are listed in alphabetical order by default. If you want an item in a list to be positioned in that order, but according to an alternative name (sort key), use the following syntax:
[[Category:category name|sort key]]
For example, to have Albert Einstein article sorted as "Einstein, Albert". You would type the following:
Singular or plural
It is not obvious whether a page like Amsterdam should be in category City (a description of a member of the category) or Cities (a description of the category as a set of pages). A convention for using one or the other is useful and should be codified. (Here is Wikipedia's convention). Sticking to the convention will help to avoid having both categories, with part of the pages in one, and part in the other.
Moving a category page
The only way to move a category page is to manually change all category tags that link to the category, and copy the editable part. There is no automatic way to move a category page in the way one moves an article page.