symmetric key system: uses (essentially) the same key for encryption and decryption
private key system: Alice and Bob have to privately agree on a secret key before they can send messages
asymmetric key system: uses essentially different keys for encryption and decryption
public key system: any system where Alice and Bob do not have to privately agree on a key before messages can be sent
key exchange or key agreement system: a public key system which lets Alice and Bob agree on a secret key by having a public conversation
trap door system: an asymmetric public key system where the encryption key is public but the decryption key is private
Merkle's puzzles is a key exchange system.