How do I send private messages to another user on the website? More specifically, how can I send private messages to a moderator on the website?
If you want to contact a specific moderator: check their profile. Some moderators provide their email address in the profile, and indicate it may be used for moderation-related inquiries. If they don't, then this is not an option.
If you want to contact some moderator, you can use a free-form flag on any relevant post (or, if no such post exists, on your own post). An alternative is the moderators chatroom for maths.SE.
SE does not provide any private messaging system for contacting non-moderators.