The GlobalPreferences extension allows user preferences to be set for all wikis in a wiki family (also known as a wiki farm). This means that users can change their preferences for all linked wikis without having to visit each in turn. Local exceptions can be set for individual wikis.
Global preferences page
The extension adds a second preferences page, accessed from the User profile tab of Special:Preferences.
This GlobalPreferences page mirrors the normal preferences page (with some items removed because they are not applicable globally) and for each preference it adds a new checkbox with which the user can select that preference to apply globally.
The GlobalPreferences page is present on all wikis in the family, and changes made via any wiki are mirrored everywhere else.
If a wiki has preferences that are only present on some wikis (for example, for an extension that is not installed everywhere) then those preferences must be set globally from one of the wikis that does have those preferences. They'll not appear on other GlobalPreferences page, and they'll not be modified when those other GlobalPreferences pages are used. Once a preference has been selected to be global, it will no longer be modifiable from the normal preferences page.
If a preference is set to be global, it will be disabled on the normal preferences page, but a new checkbox will be added under it with which an exception can be made for just the current wiki.
When you view that preference on the GlobalPreferences page, there will be a message below it indicating that there is a local exception in place. The value of the global preference will also often not be the one that's actually in effect (for example, Welsh is not the active language in the screenshot below, even though it's the active global preference).
Note that local exceptions are only visible on the wiki on which they're in effect. If you go to another wiki, there is no indication that a local exception exists elsewhere.
Removing global preferences
The global preferences page has a link, just like the normal preferences page, to remove all preferences. This will (after prompting for confirmation) delete all global preferences and all preferences will be restored to either the local value a user has set, or the site default.
Note that this deletes global values even for preferences that are not present on the current wiki. For example, if you set a global preference for an extension that's only on one wiki, and then remove global preferences via another wiki, when you go back to the first wiki's global preferences page you will not have any global preferences set. Removing global preferences does not remove local exceptions, but rather just disables them (i.e. local preferences or site defaults will be in effect). If a preference is made global again, its local exception will be reinstated.
If you encounter any problems using the GlobalPreferences extension, you can ask for help on the talk page. If you have found a bug or want to suggest a new feature, please open a ticket on Phabricator and tag it with 'MediaWiki-extensions-GlobalPreferences'.