Cold storage (also known as a Cold wallet) refers to keeping your bitcoin completely offline. This is the opposite of 'Hot wallet' where your bitcoin funds are kept online, typically in an online wallet or online Bitcoin exchange.
Cold storage is often a necessary security precaution. Since Bitcoin is a digital asset, keeping them online increases your risk or attack surface for having your bitcoin stolen from an online third-party service. By keeping them in cold storage, your attack surface is diminished.
There are different methods to storing your bitcoin offline in cold storage, such as using a paper wallet or using an offline bitcoin hardware wallet.