Question 1
A private key is a large random number that your computer generates. The public key is derived from the private key using a special mathematical formula. You can figure out the public key by using the private key. However, you cannot figure out the private key by using the public key. It is a one-way street. You do not need to keep your public key secret. However, it is imperative that you keep your private key secret. If somebody gets hold of your private key they can read your encrypted messages or take money from your account.
Question 2
Case 1: Encryption: Imagine person A and person B want to send each other messages across a network that is not secure. They do not want people to read their messages. Person A uses their computer to generate a private key. Person A then uses that private key to generate a public key. Person B then writes a message. Person B then uses person B’s public key to encrypt that message. At this point the message becomes unreadable, only person B knows what that message is. Nobody else can read it. Person A then uses their private key to decrypt the message and make it readable again.
Case 2: Digital signatures: Imagine that person B receives a message. The sender of the message appears to be person A. However, person B wants to be certain that the message they have received was actually from person A. Here is what happens. Person A uses their computer to generate a private key. Person A then uses a mathematical operation with their private key to digitally sign the message. Then person A sends this message along with the attached digital signature to person B. When person B receives the message they can see person A’s digital signature as well as person A’s public key. Person B can then use person A’s digital signature and person A’s public key to mathematically verify that the digital signature came from person A’s private key.