The word "wallet" is a little misleading, and clearing that up makes everything else click into place.
Your wallet doesn't store coins. It stores keys. Your crypto never leaves the blockchain, that shared record we talked about earlier. What a wallet actually holds are the private keys that prove you own it, and let you move it. So a better mental image isn't a wallet full of cash. It's a keychain. The money is in the vault (the blockchain); your wallet holds the key that opens your part of it.
This is why the earlier lesson matters so much: whoever holds the keys controls the crypto. The wallet is just where those keys live.
Wallets differ in two ways. The first is who holds the keys.
Custodial vs non-custodial (who holds the keys)
A custodial wallet means a third party holds your keys for you. A crypto exchange account is the typical example. When you sign up for something like Coinbase, you don't get a seed phrase; the exchange manages the keys on your behalf and you log in like a banking app. Convenient, familiar, and if you forget your password you can usually recover the account. The trade-off is control: you effectively hold an IOU from the exchange, and if it's hacked, freezes your account, or goes bankrupt, your funds are at risk. Hence the phrase you'll hear a lot: not your keys, not your coins.
A non-custodial wallet (also called self-custody) means you hold the keys yourself. When you create one, it gives you that seed phrase, and you alone are responsible for it. The upside is true ownership, no company standing between you and your crypto. The downside is full responsibility: lose the seed phrase and no one can help you recover it.
The second difference is whether the wallet is online or not.
Hot vs cold (online or offline)
A hot wallet is connected to the internet, usually a phone or browser app. It's convenient for everyday use, but being online means more exposure to attacks.
A cold wallet keeps the keys offline, usually a small hardware device that looks like a USB stick (Ledger and Trezor are common). Harder to use day to day, but far safer for holding larger amounts long term.
These two pairs mix and match. An exchange account is a hot, custodial wallet. A hardware device is a cold, non-custodial one. A common approach is to use both: a hot wallet with a little crypto for everyday use, and a cold one for the bulk you're holding long term, the same way you keep some cash in your pocket and the rest in a safe.
The takeaway
A wallet holds keys, not coins. "Custodial vs non-custodial" is about who holds those keys (a company, or you). "Hot vs cold" is about whether they're online or offline. There's no single right answer, just trade-offs between convenience and control, the same theme running through all of crypto. And whichever you choose, the rule from the next lesson never changes: your seed phrase is everything, so guard it.