IN PLAIN ENGLISH:
Cryptocurrency wallets, much like the wallets you carry in your pocket or purse, are where you store your currency and send or receive it.
The details:
Cryptocurrency wallets generally fit into one of two main categories: hot wallets and cold wallets.
A hot wallet means it is connected to the internet, which is less secure but easier to do transactions with. This type of wallet is ideal for carrying a small amount of coin and using for everyday transactions. An example of this would be a mobile wallet, where you can do transactions directly from your smart phone. Most cryptocurrencies offer official wallets that can be used.
A cold wallet is offline, making it very secure but a little less convenient to use. This is where you should store the bulk of your crypto for safe-keeping longterm because it is not accessible to be hacked or stolen online. Examples of cold wallets are hardware wallets, such as the Ledger Nano S, and paper wallets.
In order to send or receive funds through the blockchain, private and public keys are used. These are the cryptographic identifiers that reference your wallet. Public keys are what you use to receive money. Also referred to as an address, they can be shared with anyone because money can only be deposited to your account using them, so there is no risk of your funds being stolen. Private keys, on the other hand, should never be shared with anyone! The private keys are used to send money and prove ownership of your wallet. If someone got ahold of your private keys, they essentially have control of your wallet and can steal your crypto.
Hardware wallets safely store your private keys on a physical device, such as the Ledger Nano S, which is not connected to the internet, and is password protected. If the hardware wallet was ever lost or stolen, nobody would be able to access your wallet without knowing the PIN code, and you could restore your account on another hardware wallet without much trouble.
Paper wallets store your private keys on a physical piece of paper, which is completely offline. This makes it safe from hackers, but less secure than a hardware wallet since it is not password protected. If someone stole the piece of paper in-person, your account would be compromised.