If you need the archive index for a custom taxonomy, you would use taxonomy-{taxonomy}.php template file. So, if you’ve created a custom taxonomy, gallery-exhibitions, the archive index for this taxonomy would use the taxonomy-gallery-exhibitions.php template file.
If you need the archive index for terms for your custom taxonomy, you would use taxonomy-{taxonomy}-{term}.php template file. So, if you’ve got a term, future-exhibitions, for your custom taxonomy, the archive index for this term would be taxonomy-gallery-exhibitions-future-exhibitions.php.
(Note: I’m not sure how hyphenated taxonomy and term slugs will impact the template hierarchy; it shouldn’t, but if you run into problems, you might try using single-word terms for taxonomy and term.)