On a technical basis, there are a few things that the "subscribe" button might do:
Sync+Archive - Set your node to continuously download the content from this site, archive it locally f0r offline use, and provide redundancy to this community. Supported on the desktop app only.
Email Subscription - Tell your notification server that you want emails about some content.
Public Relationship Data (Contact) - Sign and publish some content that declares to the world that you have some kind of relationship to the subscription target. Currently we model this using Contacts, but it could change
From the UX side, it is clear that the subscribe button should do all of these things by default, but the user may eventually select that they only want some of this.
Contacts
When you are subscribed, you always will have a public contact for the thing you are subscribed to. When you unsubscribe, you delete the contact.
Your site server and gateway will be responsible for syncing your contact list to new devices, which will
Emails
If you have an email notification server set up, you will immediately enable notifications. The email notification server will only keep track of the target accounts where you want notifications. So it is possible to be publicly "subscribed" but without receiving email notifications.
Sync+Archive
The local subscriptions will follow the combination of Email Notifs plus all Contacts, for every account on your node/vault.