Bitcoin's are stored in a wallet.
Your Bitcoin wallet lets you transact with the world. It gives you a Bitcoin address that can receive coins from other users and lets you send those coins.
Similar to email, you can receive Bitcoins when you are offline, and all wallets are compatible with each other.
Wallets are Software, Mobile and Web based. They can also be combination of several types.
Some examples of wallets are:
Coinbase and Blockchain.info are both popular Web and Mobile wallets.
Other options are available here.
