A typical forum setup will allow you to organize topics by forum groups, forums and possibly sub forums.  In most cases, this 3-tier approach will be sufficient for keeping topics organized and easy to find.  Adding tags to each topic is another way to help classify the subject matter of a topic in a particular forum.  

If you aren't familiar with tags then think of being able to add a list of user defined keywords to each topic.  This means that the user can add keywords that they believe are relevant to the topic they are creating.  This is a great way to allow access to similar content that might be shared across multiple forums.  One disadvantage to using tags is that it is up to the user to create appropriate and relevant tags that not only match the subject matter of their topic, but also provide proper context to the overall forums.  A similar way to classify content with a little more structure is to use categories.  

Categories are defined by the forum admin and are identical in concept to tags.  The advantage to using categories is that topic authors can select from a predefined list of categories to further define their content.   In some situations you may want to use tags and categories.  Categories can be used as your primary classification and allow the use of tags to further classify the content in more detail.

Starting with Active Forums 4.3, the ability to Tag and Categorize content is a security option.  This allows admins to better control who can tag and classify content.