What Is a Crypto Wallet and How Does It Work?

A crypto wallet does not hold your coins. The coins never leave the blockchain, the shared public record that every computer on the network keeps a copy of. A wallet holds the secret keys that prove those entries are yours, and it signs payments on your behalf. The name is misleading: it is a keyring, not a purse.
What is actually inside a wallet?
When you set up MetaMask or a Ledger, it creates a seed phrase, usually twelve words. From that phrase the wallet works out your private keys, and from each private key it works out a public address you can share. One phrase can produce endless accounts across several blockchains, which is why the same twelve words can restore your Bitcoin, your Ethereum and the tokens sitting alongside them. The wallet app is really just a viewer and a signing tool over that one secret.
Hot wallet or cold wallet?
A hot wallet is software on a device that is online: MetaMask in your browser, Trust Wallet or Phantom on your phone. It is free, instant and convenient, and it is only as safe as the device it sits on. A cold wallet, such as a Ledger or Trezor, keeps the key on a small offline device and asks you to confirm every payment on its own screen, so malware on your laptop cannot sign anything without you pressing the button. Many people run both, the way you keep cash in a pocket and savings in a safe.
How is that different from an exchange account?
An account on Coinbase or Binance is not a wallet you control. The exchange holds the keys and owes you a balance, much like a bank. That buys you a password reset and a support line, and it costs you control: exchanges have frozen accounts, and some have collapsed with customer coins inside. Hence the old line, not your keys, not your coins. Neither option is automatically right. A beginner with a small amount may be safer on a reputable exchange than with a phrase they will misplace.
If you are setting up your first wallet, do it in this order: install the app from the official site, write the phrase on paper before funding anything, send a tiny test amount to yourself, and only then move a real sum. Read next: what a seed phrase is and how software wallets work.