Symmetric encryption is where a secret is shared between the sender and receiver of data. This shared secret is either the password that is used to both encrypt and decrypt a message, or it is used to generate passwords for both encryption and decryption.
If the secret is discovered by anyone else then that person will be able to decrypt and understand each message transmitted
Asymmetric encryption uses a a pair of numbers that work as public and private keys. The public key can be shared with anyone and the private key is kept secret. The public key is used to encrypt a message that can only be decrypted with the matching private key.
This is more secure than symmetrical encryption because two people sending secure messages don't need to first share a secret key. They can each encrypt a message using each other's public key. The only person able to decrypt this message will be the person who has the matching private key.