Someone I met told me about this enterprise that offered him a newsletter system that would enable him to send newsletters to his clients and tell him which clients actually read it and which, for example, sent it to the trash, marked it as spam, etc. Is this really possible? If it is, how do they do it?
It is possible when you make use of return receipts.
Not all email clients support these features, and most of them ask the end-user whether they want to send such notifications.
For a more technical overview of the possibilities, read the RFCs mentioned on the wiki page:
Delivery Status Notifications:
Message Disposition Notifications
An alternative method that is generally frowned upon is to make use of embedded images in the email, with a unique URL for each recipient.
When the image URL is requested on the web server, the sender knows the email has been opened.
This is why many mail clients block images by default, and the user has to specifically allow images to be shown in emails.
most of these systems (and there are many) use images or external links to track who/when/where the mail was opened.
technically, the most they can tell you is when something was opened (plenty of clients block images and alternative tracking methods, so it's not accurate), they can guess at when an email was forwarded (and then opened) and provide a vague deductive guess at how many went unread (and of course the obvious stuff like tracking click-throughs on links, unsubscribes, MTA related information like bounces etc.)
There would be no way of knowing if a mail was trashed or marked as spam within a client.
Mailchimp has a pretty good "these are all the things we report on" overview - they are a pretty good base to understand the bulk of mass mail systems and list managers.
There are some systems that can tell you when somebody has for sure read (or rather at least opened) your message, but there are none that can say for sure that you have not. It is possible to view letters without opening images or giving off any other "signals" that are used to identify an open. It is not possibly to accurately track messages sent to trash or spam as this happens on proprietary systems outside the control or monitoring scope of anyone sending emails.