Cold storage is a way of holding cryptocurrency private keys offline in a physical device. It differs from hot storage in that the cold storage device is not connected to the internet at any time, and is also unpowered except when necessary. You are the sole owner of your private keys at all times with a cold storage wallet, and nobody can remotely access them on the cold storage device.
Hot storage refers to crypto assets that are stored online in a digital wallet, typically with a third-party acting as custodian, such as on an exchange. Because the hot storage is online, hackers may have an easier time hacking into hot wallets as opposed to cold wallets. Furthermore, you do not control your private keys in an exchange wallet, since the exchange acts as the custodian. This means that if the exchange gets hacked or goes down, you may lose access to your crypto.
Hot storage also includes wallet apps you can download on your phone or computer which do not require any external hardware devices to operate. They use your phone's or computer's hardware to store your private keys, and these devices can be hacked as they are exposed to the internet. This is not as secure as storing your private keys offline in a cold wallet such as the Arculus Cold Storage Wallet™.