We store fiat cash in a physical wallet, and cryptocurrency works in a similar way. Wallets store the private keys that you need to access an address and spend your 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 wallets.
What wallet software you use is entirely up to you, and there are a bunch of options available. If you are using the Counterparty platform then you probably want to use counterwallet.io. If you are using the Dogeparty platform then you probably want to use wallet.dogeparty.io