Categories of Controlled Vocabularies
As defined in Classification of Controlled Vocabularies, vocabularies can be defined by their structure and form. There are three broad categories of controlled vocabularies: flat, multilevel, and relational (also called a relationship list). Within these three categories, there are a variety of types of controlled vocabularies. A type is a simplified name for vocabularies further classified by function. (Click on image to enlarge.)

Flat Vocabularies
Authority File * Glossary * Dictionary * Gazetteer * Code List
All flat vocabularies contain a label and a value. Some flat vocabularies build upon this foundation by adding a definition or additional information about each value. No relationships are established, no hierarchies are set up, and no complicated matrices are necessary. Flat vocabularies are sets of two to four pieces of information: a label, a value, and possibly a definition and additional information.
Multilevel Vocabularies
A multilevel vocabulary is essentially a way to group terms into classes with hierarchy. A classification tells more about the terms by placing them into well thought-out subcategories.
Think of a classification as a tree with a trunk, limbs, branches, and leaves. If you look at an individual leaf on the tree, you can backtrack to the branch, to the limb, and eventually to the trunk.
In a multilevel vocabulary, you can examine in which subcategory a term belongs, and you can examine the relationships between subcategories as well. In some multilevel vocabularies, the only connection between the subcategories is a broader than/narrower than comparison (taxonomy). In others, you can compare similar categories across broader categories (subject heading).
Relational Vocabularies
Thesaurus * Semantic Network * Ontology
Relational Vocabularies, also called relationship lists, contain a mechanism to connect terms. The relations are described by various standards and protocols, such as for thesauri in the ANSI/NISO
Z39.19 - 2005 standard, including broader than/narrower than, used for, and related.
The principles of a relationship list can be illustrated by seashells. You might find one on the San Diego coast that looks exactly like one in Monterey Bay, except the shell in Monterey Bay has been degraded by extreme waves and the San Diego shell has not. These two are probably related (similar to broader than). Or, perhaps you see a shell on the Oregon coast, and it looks similar to a shell in Woods Hole, Massachusetts. However, the two aren't exactly the same, because they're different colors. There is still a relationship to be defined (similar to related). Perhaps you find a message in a bottle in Maryland. The bottle came from Europe, and there's a seashell inside. In this case, you would want to relate the shell to other European shells (similar to used for).