A symmetric cipher is an algorithm that uses the same secret (private key) for both encryption and decryption. An asymmetric cipher is an algorithm that uses two different secrets, a public key for encryption, and a private key for decryption.
In English, basically the job of cryptography algorithms are to make text or files jumbled so it can't be read except by the desired recipients. Different algorithms use different techniques for doing this such as switching the order of letters or substituting them with a different letter, and the secret provides the exact instructions on how to do that.
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