A private key is a randomly generated number created by a computer. This number serves as the access code for a user into a blockchain (or similar) network. Only the user should know the private key for security of access to the information held within that address (key). This key provides the user privacy and security of personal information, or in the case of cryptocurrency - value.
A public key is mathematically derived by a computer to relate to the private key. The public key can then be shared across an open network to other users in order to have information (or value) sent to it. Access to the public key is then only granted to the user that holds the corresponding private key.
The two use cases for public key cryptography are:
a) Information encryption (privacy/security of info transfer on a network).
b) Value encryption (privacy/security of value transfer on a network [Bitcoin etc].