A wallet is a piece of software which generates and stores public and private keys, allowing users to send, receive, and store bitcoin. Wallets can come in the form of mobile apps, desktop apps, or hardware devices. Wallet is a general term for any of these implementations, and wallets differ greatly on security and privacy.
Wallets are most useful for abstracting the complexities of Bitcoin and cryptography away from the user, so that they can transact and store their bitcoin in a simple manner. Wallets will show users their balances, their transaction history, and their addresses, which users can publish in order to receive transactions.
Selecting a wallet with quality security and privacy practices is important, as is backing up the wallet in case you lose or break your device.