1. Terminology: "Taxonomy" and "Vocabulary" mean the same thing -- a collection of terms used for tagging nodes. Similarly, "Term", "Tag", and "Category" are synonyms, and "to tag" and "to categorize" mean the same action. Drupal sometimes uses these terms interchangeably, which could erroneously cause one to believe they refer to different features. In fact, they are all the same.
2. A *free-tagging* taxonomy is one where content authors can add new terms to the vocabulary "on the fly", e.g. while editing a post, you just type in the terms you want and they are automatically added to the vocabulary. Free-tagging is the easiest way to tag things, and the use of auto-completion text entry can enable the vocabulary to be extremely large without affecting the ease of selecting the right tag. However, free-tagging can lead to inconsistent use of tags since there is no oversight by an expert to ensure that tags make sense and are created in the correct vocabulary. The alternative is the *managed* taxonomy where the tags are exclusively controlled by an administrator. Managed taxonomies are best used with small vocabularies that cover a finite number of terms which are known in advance, or for vocabularies constrained by some other requirement that needs qualified administrator to enforce.
3. A taxonomy can also have one of three different structures: flat, hierarchical, and graphical (multi-hierarchical). However for various reasons, it turns out that flat is the best choice for nearly all situations. Studies have found that hierarchical structures are often bad for user-interface design because sub-terms can be hidden from view and the user may not know which place to find something. This problem gets worse as the depth of the hierarchy increases, and so we recommend that vocabularies be kept either completely flat or use minimal hierarchy when absolutely necessary.
