Cold storage describes any method of storing data which does not connect to the internet or any other devices. A cold wallet is a form of cold storage often used by bitcoiners to protect bitcoin which they do not plan on moving frequently.
A wallet is considered a cold wallet if the private keys are kept on a device isolated from the internet. However, the public keys of a cold wallet can still be used on a separate device connected to the internet. This allows users to receive funds directly to a cold wallet without sacrificing any security.
Cold storage is more secure than hot storage because almost all malware is introduced to devices through the internet. However, this benefit comes with inferior convenience. Thus, cold storage should be used for large amounts of bitcoin that are not intended for day-to-day use.