URL alias to call function
Okay, here is a proper answer to a tricky question.
Documentation
First of all, you should read the following two documents, if you haven't already:

https://api.drupal.org/api/drupal/modules%21system%21system.api.php/function/hook_menu/7.x
https://www.drupal.org/node/224170

Simple Wildcards
By using % as a path wildcard you are instructing the menu_router matching algorithm to use the path argument as it is (used mostly for numerical arguments). So, following from your comment, yes, it does have a significance. If you write your code like this:
$items['admin/config/mapper/%/edit'] = array(
  'title'            =&gt; 'Mapper edit mapping',
  'description'      =&gt; 'Add or edit a mapping',
  'page callback'    =&gt; 'drupal_get_form',
  'page arguments'   =&gt; array('mapper_form_mappings', 3),
  'access arguments' =&gt; array('administer site configuration'),
  'type'             =&gt; MENU_LOCAL_TASK,
  'file'             =&gt; 'mapper.admin.inc',
);

you are going to get the 3rd part of your path as an argument to your mapper_form_mappings function. Exactly as it occurs in the path.
Auto-Loader Wildcards
Now, if you would use a %NAME type of wildcard, like taxonomy does, you would be triggering the 'Autoloader Wildcards' (a.k.a. 'Wildcard Loader Arguments') functionality. Drupal will try to load an object through a function called NAME_load. In the case of the %taxonomy_vocabulary_machine_name Drupal will try to load a vocabulary using the specified machine name as an argument to the taxonomy_vocabulary_machine_name_load function. The loaded object will then be passed as an argument to drupal_get_form (or any other function defined in your path definition).
Current problem
In your case, your original code:
$items['admin/config/mapper/%tid/edit'] = array(
  'title'            =&gt; 'Mapper edit mapping',
  'description'      =&gt; 'Add or edit a mapping',
  'page callback'    =&gt; 'drupal_get_form',
  'page arguments'   =&gt; array('mapper_form_mappings', 3),
  'access arguments' =&gt; array('administer site configuration'),
  'type'             =&gt; MENU_LOCAL_TASK,
  'file'             =&gt; 'mapper.admin.inc',
);

implies there is a tid_load function that will receive the %tid argument from the URL and should return some object to drupal_get_form. This is however not working, since such a function does not exist.
Alternatives using wildcard loaders
If you want to use the 'Auto-Loader Wildcards' functionality, you should do one of the following:

Use an existing _load function, e.g. taxonomy_term_load. Your wildcard should be called %taxonomy_term in this case and the final argument passed to drupal_get_form would be an taxonomy term object.
Use a custom _load function, e.g. mapper_load. Your wildcard should be called %mapper in this case and the final argument passed to drupal_get_form would be whatever your mapper_load function returns.

Best of luck!