Tags can be used to sort and select Mailchimp subscribers.
They can be used to add attributes to a subscriber.
They are private to you and cannot be seen by the subscriber.
Each subscriber can have one or more Tags.
Some enterprises prefer to use Tags on their own, without using Groups.
Groups can be seen externally, i.e. within a Mailchimp signup form, like this one.
Tags can be used, for example, to promote an event. You could Tag, say, 100 subscribers and send them an invitation to your event via Mailchimp. If you delete the Tag from subscribers who have registered, you can then send another campaign to the people who have not yet registered.
I have seen Tags used for sales pipeline management and numerous other purposes, as they are so flexible.
Nigel Temple provides Mailchimp training services.