Authentification

In a secret key system, the sender and the receiver usethe same key to encrypt and decrypt. This is a two way confidentiality.
In a public key system, the sender, say Bob, uses the public key of the receiver, say Alice, to encrypt the message, and Alice uses her private key to decrypt the cipher to get the message. Bob is sure that only Alice can decrypt the cipher. This is a one way confidentiality.
Secure Socket Layer and IPsec offer two way confidentiality.
If Bob can decrypt it using Alice's public key, he can be sure that it was Alice who encrypt (using her private key) and send him this message. Certainly, everyone can use Alice's public to decipher the message. The point is that everyone then know for sure that no one else but Alice who sent the message.
<b>Hashing and digital signatures with private key.</b>