What is a crypto wallet?
A crypto wallet does not really store your coins. Your balance lives on the blockchain; what the wallet manages are the keys that prove it belongs to you. Your private key is what authorises access and signs transactions, which is why protecting it matters more than anything else you do.
That leads to the first real choice. With a self-custody wallet the keys are generated on your own device and stay with you, so nobody can freeze or move your funds — and nobody can recover them for you either. With a custodial wallet, such as an account on an exchange, a company holds the keys on your behalf. That is more forgiving if you lose a password, but you are trusting them with your money.
The second distinction is how connected the wallet is. A hot wallet runs on an internet-connected device — a phone or a desktop — which makes it convenient for everyday sending and receiving. A cold wallet keeps keys offline, which suits larger amounts held for longer. Plenty of people use both: a mobile wallet for day-to-day funds, something offline for savings.
For a closer look at who controls what, read custodial versus self-custodial wallets.