Symmetric-key encryption only involves one key, and you just use that one key to both encrypt and decrypt a message.
Whereas, public-key (asymmetric-key) cryptography is any cryptographic system that uses pairs of keys:
Public key,
which is a large numerical value that is used to encrypt data, and is made available to everyone through a publicly accessible repository or directory.
Private key,
which is another large numerical value that is mathematically linked to the public key.
Whatever is encrypted with a public key may only be decrypted by its corresponding private key and vice versa.
The cryptography accomplishes two functions:
Authentication,
where the public key verifies that a holder of the paired private key sent the message, and
Encryption,
where only the paired private key holder can decrypt the message encrypted with the public key.