From the interface side of things, it’s not used for anything at all.
However, internally, it’s possible to create terms that are aliases of other terms. These terms will then be in the same term group, showing that they’re really the same term.
There is no real use for it at this point. It’s there for potential future functionality and for plugins.
If you are a plugin programmer and know how WP works, then you can add an alias_of parameter to the call to wp_insert_term(). The alias_of specifies the slug of the term that the new term you’re inserting will be an alias of (and so which will share a term_group). Same parameter works on wp_update_term() too.
