With symmetric cryptography, the same key is used for both encryption and decryption. A sender and a recipient must already have a shared key that is known to both. Key distribution is a tricky problem and was the impetus for developing asymmetric cryptography.
With asymmetric crypto, two different keys are used for encryption and decryption. Every user in an asymmetric cryptosystem has both a public key and a private key. The private key is kept secret at all times, but the public key may be freely distributed.
Data encrypted with a public key may only be decrypted with the corresponding private key. So, sending a message to John requires encrypting that message with John’s public key. Only John can decrypt the message, as only John has his private key. Any data encrypted with a private key can only be decrypted with the corresponding public key. Similarly, Jane could digitally sign a message with her private key, and anyone with Jane’s public key could decrypt the signed message and verify that it was in fact Jane who sent it.
Symmetric is generally very fast and ideal for encrypting large amounts of data (e.g., an entire disk partition or database). Asymmetric is much slower and can only encrypt pieces of data that are smaller than the key size (typically 2048 bits or smaller). Thus, asymmetric crypto is generally used to encrypt symmetric encryption keys which are then used to encrypt much larger blocks of data. For digital signatures, asymmetric crypto is generally used to encrypt the hashes of messages rather than entire messages.
A cryptosystem provides for managing cryptographic keys including generation, exchange, storage, use, revocation, and replacement of the keys.