A private number is that when the caller holds back his phone number with the help of the service provider and the service provider doesn't send the Caller ID information to the recivercall receiver then it considered a "Private Number". To activate the feature ,
Open Dialer > 3 dots > Settings > Call Settings -> Sim related call settings > Additional settings > Caller ID
Note: Not all carriers allow hiding the Caller ID. Mostly it is greyed out for security purposes.
On the other hand, when your telephone exchange(service provider) is unable to register the Caller's number, then it is called then it is called "unknown".