Unless you are a project administrator you can only delete templates that were created by you. Read more
Open "Manage Templates" dialog:or go to "Checklist Templates" page:
The screenshots below were taken from Manage Templates dialog but the process is identical on Checklist Templates page.
If template to be deleted is used as the default checklist in other projects (i.e. other than the one you're currently in) you will need to confirm your intention:
Templates aren't removed straight away after you click "Delete". You can undo that operation as long as the dialog / page remains open.
If you don't undo the deletion, the template will be permanently removed after you close the dialog or navigate away from the page.