If you do not have administrator rights you can only delete your own reminders. Project administrators could delete reminders in their own projects and JIRA administrators could delete all reminders. If you do not want to receive notification from a reminder just tell the reminder owner to remove you from reminder targets. If you delete an active reminder notification will not be sent. Deleting a past reminder will not have any effect other than removing data from the database. Since an already delivered notification has nearly no overhead on the system you generally do not need to delete them.
