Public-key cryptography has been used for years in traditional computing to send encrypted messages. It uses mathematical functions close to impossible to reverse engineer, ensuring their security.
Blockchains use a system of public and private keys utilizing public-key cryptography to secure transactions.
The public key is analogous to an email address — anyone can send emails to the address but you need the password to get full access. A public key can be openly shared.
The private key is analogous to the password for an email address — it gives you full access and control over the email address. The private key should not be openly shared.
Anyone with access to both your public and private keys has full control over your blockchain “account”. If your assets are stored on an exchange, then they actually fully control your blockchain “account”, and you just hold a “receipt” for what they owe you.
Private keys are being used in interesting ways, such as creating a “brain wallet”. If you can memorize the mnemonic phrase representing your private key, then you can “store” an unlimited amount of digital assets in your head.
A brain wallet is immune to most online and offline attacks. It is probably the only technology that allows individuals to store and move with value completely in their minds. You could walk through a border with a billion dollars in your head, and no one would know.