What is a wallet and why "not your keys, not your coins"
A crypto wallet doesn't store your coins: it stores the private keys that give you control over them. Understanding this difference is the foundation of crypto security, and what the famous phrase "not your keys, not your coins" sums up.
What a wallet really is
Your crypto lives on the blockchain, not inside an app. What a wallet stores is your private key: a kind of master password that proves those coins are yours and lets you move them. Whoever has the private key controls the coins. Full stop.
Custody vs self-custody
There are two models. In custody, a third party (an exchange like Binance or Coinbase) holds your keys for you: convenient, but you trust them. In self-custody, you hold your own keys (in a wallet you control): more responsibility, but total control. The difference is who really holds the keys.
"Not your keys, not your coins"
This phrase sums it up: if you don't have your keys (because an exchange holds them), you don't actually have full control of your coins, just the exchange's promise to return them. If that exchange goes bankrupt, is hacked or freezes withdrawals, your coins are at its mercy. It has happened (FTX is the famous example).
Types of self-custody wallet
Hot wallets: apps connected to the internet, convenient for daily use but more exposed. Cold wallets: offline physical devices (hardware wallets), much safer for storing significant amounts long term. The usual rule: a little in the hot one to trade, the bulk in the cold one.
The seed phrase
When you create a self-custody wallet you get a 12- or 24-word phrase. THAT phrase IS your wallet: whoever has it has your coins. Never share it, never upload it to the internet, never store it in a photo. If you lose it, you lose access forever; if it's stolen, everything is stolen.
Where to start
If you hold small amounts to trade, a serious exchange is acceptable for convenience. As soon as you accumulate an amount it would hurt to lose, learn to use self-custody (ideally a hardware wallet) and store your seed phrase on paper, somewhere safe and offline.