A wallet does not contain cryptocurrency. The coins exist on the blockchain. A wallet holds the private key that authorises moving them, and every distinction between wallet types is about where that key lives.
Custodial versus non-custodial
Custodial means someone else holds the key – an exchange, usually. You have an account with them, and they can freeze it, lose it, or be compelled to hand it over. You also have password recovery and someone to contact.
Non-custodial means you hold the key. Nobody can freeze your funds and nobody can help if you lose the key.
Hot versus cold
Hot means connected to the internet – a phone or browser wallet. Convenient, and exposed to anything that compromises the device.
Cold means the key never touches an online device. A hardware wallet signs transactions internally and only the signature leaves. Far more secure and less convenient.
The recovery phrase
Twelve or twenty-four words that regenerate the key. Anyone with those words has the funds. Losing them with no backup means permanent loss.
Written on paper or metal, stored somewhere secure and not photographed. Never typed into anything that asks for it – that request is the single most common theft method.
What suits what
Small amounts you transact with: a hot wallet or a reputable exchange, treated like cash in a pocket.
Amounts you would be hurt to lose: a hardware wallet with the phrase backed up in two physical locations.
The caveat
General information. Nothing here is advice about buying or holding anything.