Offline storage in cryptocurrency terms means your private keys, and the coins associated with them, are not accessible to the internet – they are “offline.” This is considered to be far safer as any coins that are online are able to be accessed by others, particularly hackers who might want to steal them. This makes it especially important to use offline storage whenever possible.
The only time your coins should be online is to make a transaction. There are three excellent means of offline storage. One is paper wallets, although this still exposes your coins to the internet when transactions are made. Another is to keep your coins in a wallet that resides on a computer or drive not connected to the internet.
This has the same weakness as paper wallets however, in that your coins are exposed any time you make a transaction. The third and preferable method is to make use of a hardware wallet such as the Trezor or Ledger Nano S, which keeps your private keys offline even during transactions.