What is a Cryptocurrency Wallet?
Similar to a traditional wallet, a cryptocurrency wallet allows you to store, send, and receive currency. A cryptocurrency wallet, furthermore just called a wallet, consists of a public address and a private key.
A wallet is a small, flat case that can be used to carry such personal items as cash, credit cards, and identification documents (driver's license, identification card, club card, etc.), photographs, transit pass, gift cards, business cards and other paper or laminated cards. Wallets are generally made of leather or fabrics, and they are usually pocket-sized but not always foldable. - Wikipedia
The public address is your wallet address. You can give this to anyone. This allows other people to send you currency.
The private key is, well, private and should not be shared with anyone. The private key allows you to send bitcoins to other people.
When making a transaction, the recipient's public key is combined with your private key. This combination makes the transaction possible.
Different Types of Wallets
There many different wallets available. To make things easier, let's group wallets into 6 categories: Hardware Wallets, Mobile Wallets, Desktop Wallets, Web-based Wallets, Exchange Wallets, and Paper Wallets. Within each of these categories, there are wallets specific to different types of currency e.g. Bitcoin, Ethereum, Neo, etc.