Wallet
Wallet is what stores and manages the information of the cryptocurrency. It contains both public key and private key, each of which has different purposes. In short, public key works like an account number and private key like the password for that account. In the meat world the wallet has cash inside, but wallet for cryptocurrency does not exactly works like that. Cryptocurrency, by its nature, are actually stored in blockchain, not its digital wallet. Digital wallet, as said in the first sentence, contains the information of it. There can be two kinds of wallets, Hot Wallet and Cold Wallet, depending on their connection to the network.
Hot Wallet
Hot Wallet is one that is connected to the network and most people refer to as the "wallet." In case of hot wallet, private key which is a part to be protected with more attention is also connected to the entwork. Although not as open as anyone can easily find, possibility of theft via hacking is always alive for hot wallets.
Cold Wallet
Cold Wallet is one that is not connected to the network. It comes in form of paper, USB, or other kinds of hardware that are disconnected to the network. Since hot wallet is exposed to online hacking, cold wallet is considered a safer way to manage one's cryptocurrency though obviously less convenient to use. Cold Wallet is also called Cold Storage.
Paper Wallet
Paper Wallet is a kind of Cold Wallet, which is a piece of paper where information of cryptocurrency, like public and private keys, is written. While it is used as a secure way to store the information, you need to be careful not to expose the private key written on the paper and moreover not to lose the paper wallet itself