A Bitcoin wallet is a combination of keys that hold bitcoin. Bitcoin wallets interact with the Bitcoin blockchain. They monitor transactions and update wallet balances accordingly.
There are various different types of wallets, some are connected to the Internet while others are not. Different wallets are used for different purposes, such as accessibility or privacy. The crucial difference amongst wallets relates to how their private key is stored.
The private key acts as a password to sign outgoing transactions and whoever knows it can control a wallet's bitcoin.
The public key and its associated bitcoin are visible on the blockchain. The public key can generate new Bitcoin addresses for receiving bitcoin. The creation of new receiving addresses is for privacy reasons, however, the coins are still linked to the same public key.