Moderators should lock/unlock threads (at their discretion, with or without notice), move off-topic threads to the right places (at their discretion, with or without notice), alert members who break the rules of the rules, alert administrators of concerns, etc.
Moderators should not lock threads simply because they're old.
Moderators should generally lock threads for authors who request for them to be locked.
If a thread or post is concerning for any reason in accordance with administrator guidelines, and locking or moving it is not the solution, a moderator should move it to the Moderated forum for review, where any number of things may happen to it. Moderators cannot delete threads (so, they should move them to the Moderated forum instead, where an administrator may or may not delete them).
Moderators should not unlock, nor unpin threads that administrators have locked or pinned, without permission.
Moderators can ban and unban users. You may appeal to be unbanned, but if you have not changed your ways, don't count on it. Make your appeal to an administrator. Being offended by the one who banned you will not help you. You are most likely to be reinstated if you are friendly and humble.
Banned users can still log in, and they can still email other users, unless their email privileges have been revoked by an administrator. However, they cannot post.
Moderators should not use such as a three strikes policy, for banning. Each case should be considered intelligently.
Moderators may pin threads, but should do so sparingly, and according to administrator guidelines.
Emailing users is not part of a moderator's job, at all (although administrators may do it). Moderators should do all moderator-related communications with members within the forums themselves (however, engaging with members about the moderation after having performed it is generally discouraged, beyond notification of the fact). If you receive an email from a moderator, treat it as if a regular user is emailing you. They have no authority in emails, nor do they off-site.
Please do not share emails from moderators with administrators. We have no way of knowing what you just made up and what is true. (And besides, their emails are the same as those of regular users.) You are certainly free to ignore emails from moderators, and filter them out, if you so desire.
Moderators do not automatically have access to your email address like administrators do.
If you need to talk to a moderator or administrator, where possible, please do it in the Support forum.