A bitcoin wallet is a computer, smartphone app, or hardware device that stores private keys that are necessary to sign a transaction and send bitcoin.
A bitcoin wallet also generates public keys from those private keys to receive bitcoin. Bitcoin wallets have varying degrees of security and user-friendliness, so choosing your bitcoin wallet is very important.
Typically, a backup of this private key is made during the setup process so that funds can be restored if something happens to the wallet itself. This backup is typically a twelve or twenty-four seed phrase.
The least secure way to store your bitcoin is in hosted wallets provided by companies online, like exchanges or browser-based apps. With hosted wallets, third-parties control the private keys and it requires trust that this third-party will always act in your best interest.
A more secure way to store your bitcoin is by using a dedicated hardware device. This hardware device keeps the private key disconnected from the internet at all times, and is less prone to attack.
The most secure way to store your bitcoin is by using multi-signature, or "multisig," wallets that require two or more keys to send bitcoin. This solution allows you to separate the keys from each other geographically. This greatly reduces the risk of theft or disaster.
Ideally, you should use a combination of hardware devices and multisig to secure your bitcoin. Apps like Casa make this easy.