HI John tricky issues for sure.
Lets say you want to send a message to me Peter.
How do I know it is from you John.
Aha lets say you have a private key you use to hash your message with and you then send that message with hash to me Peter.
I have the same private key
when I receive your message i hash it with my private key (symmetric encryption) if the hash is the same I have proof it came from you John.
YOU can read a more technical explanation here
What I have described is called MAC
You can make it more secure with HMAC.
You John have a private key and encrypt a message with it
you send encrypted message to me Peter
I have your public key and I can decrypt your message then I know message must be from you John
as you are the only one who can encrypt it with your private key. Authenticated and Confidential.
For integrity you need the hash