Categories are groups of articles on related topics. At the bottom of an article, you will see a box containing the categories to which that article has been assigned. Simply click any of these categories to go to the corresponding category page. From there you can browse other articles on the same topic, or navigate through the category tree to find other related categories.
How are categories organized?
Most categories have a number of other categories listed as subcategories. These are listed in a special section of the category page, which appears above the list of articles belonging to the category. Click on these subcategories to navigate through the category tree to find more specific groups of articles.
Similarly, you may want to navigate in the other direction, to find more general groups of articles ("parent categories"). These are listed in the box at the bottom of a category page, just as on a Wikipedia article page.
Why is an article not in the categories I would expect?
Articles are not usually placed in every category to which they logically belong. In many cases they will not be placed directly into a category if they belong to one of its subcategories. This is because otherwise categories would become too large, and the list of categories on articles too long. To find the articles you are looking for, it may be necessary to dig down. For example, you won't find Oslo listed at the category called Cities, but if you start from there and click "Cities by country", and then "Cities and towns in Norway", you'll arrive at the right place. Conversely, if you are at the Oslo article and you want to find the category of all cities, start by clicking Cities and towns in Norway and navigate up the tree to its parent categories.
What types of categories are there?
Different parts of Wikipedia use different schemes for organizing articles into categories. The main types of categories used are:
Administration categories or project categories – categories used mainly by Wikipedia's editors for project management purposes, rather than for browsing. Category:Wikipedia backlog links to many categories with articles that need to be worked on, sorted by reasons. Another common category is the stub category, which contains very short ("stub") articles that need expansion. Wikipedia:WikiProject Stub sorting/Stub types contains a giant list of stubs, sorted by topic.
Container categories – categories which only contain other categories.
Set-and-topic categories – categories which are combinations of the two above types.
Universal categories – categories used to provide a complete list of articles which are otherwise normally divided into subcategories.
How else can I navigate articles?
Of course, categories are not the only Wikipedia feature for browsing articles on particular topics. Readers most often find related articles simply by clicking the links that appear in the article they are reading. To find which articles contain links to the article you are reading, click What links here at the left-hand side of the page. For a general starting point to browsing the encyclopedia, go to Portal:Contents. Wikipedia also has many pages containing lists, which serve a similar function to categories.
Can I specify categories when searching?
The basic search box does not search category names. As explained here, you can access category search by clicking on "Advanced" under the special search box.
You can search for specific category names by prefixing the search term with the prefix Category:. For example, searching for "Category:Countries" will return a list of all categories whose names contain the word "countries" or "country".
When using the search box, it is also possible to restrict your search to articles tagged with a particular category. To do this, just add +incategory:"CategoryName" to the search string. As explained here, more than one category can be specified, making it possible to find articles in the intersection of categories.
Why might a category list not be up to date?
Sometimes, pages are not placed in categories manually by Wikipedia editors, but by means of templates, which can be used to place identical information (including category membership information) on many different pages at once. When the information on such a template is edited, the pages containing that template are updated, but not necessarily updated immediately. This means that pages might not always appear in the most current categories. However, this problem usually affects project maintenance categories rather than the categories used for browsing.
Various other temporary delays may sometimes mean that lists of category members or subcategories, or the page counts given, are not completely up to date. So if you are editing Wikipedia and find that your page hasn't yet shown up in a category or been removed from an old category, don't panic! The problem may resolve itself within minutes, but sometimes it may take longer, in some cases days, even weeks. (It may help if you make a null edit to the page.)