The taxonomy_display.api.php file contained within the taxonomy display project outlines the Drupal hooks exposed for implementation in your module. However, these hooks don't give a full scope of what you can achieve with taxonomy display as the strength of the module is the plugin system which doesn't utilize standard Drupal hooks, except to define the plugins in hook_taxonomy_display_plugins().
- Term display - How to - Customize output for the term's display.
- Associated content display - How to in works - Customize output for content referencing the term.
- Breadcrumbs - How to in works - Customize the breadcrumb for terms.
If you've written a taxonomy display plugin in a contributed module or think it would be a good candidate for inclusion in taxonomy display please open an issue with a link to your module or attach a patch file, respectively. I'll be happy to link to implementing modules and include quality plugins within the main project.
Object oriented programming notes
If you've never worked with Object Oriented Programming before there's really not anything special you need to know for setting up a basic plugin, just follow the recipes provided and tweak the methods to suit your needs — a method is just a function within a class.
In my documentation you may see something along the lines of ClassName::methodName() that just means go look at the function "methodName" in the class "ClassName".
If you're wondering why I decided to use classes for bundling the functionality for plugins in taxonomy display it's because it was the best choice for code isolation (making maintenance easier) and Drupal 7 added a very nice auto-loading mechanism for classes, so that we do not need any module includes or other non-sense that plug-able modules had to fight with in Drupal 6.