When you hide a connection, you will stay connected, but the connection will simply be hidden in your Cardbox contacts list. A connection does not know whether or not you have hidden them.
You can use this feature if you want to be connected to someone, but rarely need to open their card, and you want to keep your contacts list compact.
When a connection is hidden, their birthday is also no longer synced to the system address book, meaning it no longer shows up in the Birthdays calendar.
To find a hidden contact, go to Hidden Contacts on the Settings screen, or look them up using the Search field.