A bitcoin wallet is a computer, smartphone app, or hardware device that stores one or more private keys that are necessary to sign a transaction and send bitcoin, and 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.
A bitcoin wallet contains the digital private key(s) needed to send funds from the wallet. 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.
The least secure way to store your bitcoin is in hosted wallets provided by companies online, like exchanges or browser-based apps.
A more secure way to store your bitcoin is by using a dedicated hardware device, which is a small device that keeps the private key disconnected from the internet at all times, and is therefore less prone to attack.
The most secure way to store your bitcoin is by using multisignature, or "multisig," solutions that require two or more keys to send bitcoin. This allows you to separate your keys from each other geographically and greatly reduce the risk of theft.
Ideally, you would use a combination of hardware devices and multisig. Apps like Casa make this easy.