The encryption of data is done with a key, which is a description of how data is converted from plain text form into a cipher text form. Encryption algorithms today rely on math properties, so keys in such cases are numbers (integers). Similarly, decryption is also done with a key. Ideally, one can convert a cipher text back to its plain text form only if he or she has the decryption key.
Why do we say 'ideally'? Because data that was encrypted using a weak encryption algorithm might be attacked and revealed by a hacker, even when the hacker doesn't have the key. This is why encrypting your data with a strong encryption algorithm is critical!
A cryptosystem is a collection of cryptographic algorithms that implement a particular security service (a system for encryption and decryption). Typically, it consists of three algorithms: key generation, encryption, and decryption.
A cipher (also: cypher) is part of a cryptosystem. It consists of a pair of algorithms: one for encryption and one for decryption.