A cryptocurrency or digital wallet is simply a device or physical medium which stores your public and private keys and the keys determine the ownership. the private key also is used to sign a transaction and it provides mathematical proof that the cryptocurrency is actually there available to be bought or sold. And with the private keys it allows you to send and receive the different cryptocurrencies. No, the cryptocurrencies are not actually stored in the wallet because they're in digital form.
A digital wallet allows you to pay for items through your mobile phone app or you can use it on your desktop or laptop computer as well. In essence it gives the user the ability to make electronic transactions. The wallet has a bar-code that can be scanned for easy payment.
You still need to be safe when using a digital wallet you do not want to leave it laying around, ensure that you have strong passwords, PIN numbers, fingerprint or whatever safety measures you can and act on it to protect your cryptocurrency.
There are wallets for each specific cryptocurrency as well. There are wallets that can contain an array of different cryptocurrencies all-in-one wallet, which will give you the ability to buy sell and exchange one cryptocurrency for another and one good example is The Exodus Wallet. Exodus, allows you to have Bitcoin, Litecoin and other alt coins in the same wallet on your phone or your computer, you can exchange Bitcoin for another type of coin and vice versa. You can sell your coins and have the money sent back into your bank account.
This is the exodus wallet: