APIs are a form of plugins that are tightly associated with a module. Instead of a module providing any number of plugins, each module provides only one file for an API and this file can contain hooks that the module should invoke.

Modules support this API by implementing hook_ctools_plugin_api($module, $api). If they support the API, they return a packet of data:

function mymodule_ctools_plugin_api($module, $api) {
  if ($module == 'some module' && $api = 'some api') {
    return array(
      'version' => The minimum API version this system supports. If this API version is incompatible then the .inc file will not be loaded.
      'path' => Where to find the file. Optional; if not specified it will be the module's directory.
      'file' => an alternative version of the filename. If not specified it will be $module.$api.inc
    );
  }
}

This implementation must be in the .module file.

Modules utilizing this can invole ctools_plugin_api_include() in order to ensure all modules that support the API will have their files loaded as necessary. It's usually easiest to create a small helper function like this:

define('MYMODULE_MINIMUM_VERSION', 1);
define('MYMODULE_VERSION', 1);

function mymodule_include_api() {
  ctools_include('plugins');
  return ctools_plugin_api_include('mymodule', 'myapi', MYMODULE_MINIMUM_VERSION, MYMODULE_VERSION);
}

Using a define will ensure your use of version numbers is consistent and easy to update when you make API changes. You can then use the usual module_invoke type commands:

mymodule_include_api();
module_invoke('myhook', $data);

If you need to pass references, this construct is standard:

foreach (mymodule_include_api() as $module => $info) {
  $function = $module . '_hookname';
  // Just because they implement the API and include a file does not guarantee they implemented
  // a hook function!
  if (!function_exists($function)) {
    continue;
  }

  // Typically array_merge() is used below if data is returned.
  $result = $function($data1, $data2, $data3);
}

TODO: There needs to be a way to check API version without including anything, as a module may simply provide normal plugins and versioning could still matter.