A Bitcoin wallet does not actually hold coins. Bitcoin lives on the blockchain as UTXOs. A wallet holds the keys that can spend them, and it gives you a way to receive, send and see your balance.
What a wallet does
It creates and stores private keys, usually from a seed phrase. It gives you fresh addresses to receive payments. It tracks which coins belong to you, by querying a node. And when you pay someone, it builds the transaction, chooses a fee, signs it with a digital signature and broadcasts it.
Types of wallets
Custodial: an exchange or app holds the keys for you. It is convenient, but you are trusting them. If they fail or freeze your account, your coins are at risk.
Mobile and desktop (hot) wallets: you hold the keys on an internet-connected device. Good for spending money.
Hardware (cold) wallets: small devices that keep keys offline and sign transactions internally. Even if your computer has malware, the key does not leave the device.
Multisig wallets: require several keys to spend. See multisig.
Lightning wallets: manage payment channels for instant small payments on the Lightning Network.
Choosing one
A common approach is to treat it like real-world money: a hot wallet as your pocket wallet for small amounts, and a hardware wallet or multisig setup as your safe for savings. Look for open-source software with a good track record. Ideally, connect it to your own full node for privacy.
Whatever you pick, the backup is what matters. If your seed phrase is safe, your coins can be recovered in another compatible wallet. That is the freedom of self-custody.