Categories are a useful way to group many pages that share a particular subject. The full list of categories currently being used can be found at Special:Categories. Categories can have subcategories.
How categories work
To add an article to a category, for example John Doe's Bakery to Category:Bakeries, just add [[Category:Bakeries]] somewhere in John Doe's Bakery. For ease of editing, it's normally placed at the very bottom, separated from the above content by one or two blank lines.
This will add the following to the bottom of the article:
Clicking on "Bakeries" will take you to the category page, where is listed all the articles labeled in that category, including the one you just added, John Doe's Bakery.
Spiffing up category pages
If you see a category link that is red, and you know what the category is about, feel free to add a short description of what goes in that category to the category page. For example, at Category:Lumps of green putty, one might add the following description: "This category includes small masses of gelatinous substances of a greenish hue. The article associated with this category is Lumps of Green Putty." Once a category has any content at all in it, the link is now blue!
Most categories can be considered subcategories of broader categories. For example, Category:Lumps of green putty is a subset of Category:Lumps of putty, Category:Green things, and Category:Useless categories. Subcategories are helpful when a category gets too many items to be as useful. To make a category a subcategory, simply treat the category page like any article, adding [[Category:Lumps of putty]] (or something more real) to it.