A wallet is a set of private keys that gives you access to the control of certain coins on the blockchain ledger. When you spend a coin, control of that coin is transferred from being controlled by your keys to being controlled by someone else’s keys.
A wallet app (often referred to simply as a “wallet”) is an application which manages these keys, reports these amounts, and facilitates the transfer of amounts of coins between addresses. Some wallet apps store your keys locally; others store the keys on a remote server (which is less safe). Just remember: whoever has the keys controls the coins.
While some “custodial” services, where coins that are “yours” are held under the service provider’s keys may be trustworthy, keep in mind that many are not. Unless you hold the keys, ultimately, the only thing keeping whoever does hold the keys from transferring your coins is trust.
It is important to understand that unlike a physical wallet, which holds actual notes and coins, crypto wallets do not really “store” your crypto (in this case, Dogecoin) locally. They are storing a collection of private keys and private addresses used to access your Dogecoin, which is actually “living” on the blockchain. The blockchain is a digital ledger of transactions that is duplicated and distributed throughout an entire network of computers called nodes. You could have several wallet applications using the same private keys, and you would be able to access and spend your Dogecoin from each of these apps.
For more information about getting a wallet and the different types of wallets, check the article: How do I get a wallet?.