Bitcoin wallets store the private keys that you need to access and spend your bitcoin funds. They come in different forms, designed for different types of device. You can even use paper storage to avoid having them on a computer at all. Of course, it is very important to secure and back up your bitcoin wallet.
Web-based wallets store your private keys online. Several such online services are available, and some of them link to mobile and desktop wallets, replicating your addresses between different devices that you own.