Custody is simply whoever holds the private keys. There are 3 different types of custodian ship. Self custody, joint custody, and third party custody.
Self custody, is when you hold the private key (seed phrase) and you alone have the power to move coins in and out of the wallet via a public address. When taking self custody be sure the wallet gives you your keys and doesn't just store it on the device. Self custody wallets fall into 2 categories, hot wallets and cold wallets. Hot wallets are constantly connected to the internet. This includes web wallets, browser extension wallets, desktop wallets, and mobile wallets. These hot wallets are relatively secure but since they are online they theatrically can be hacked either directly or indirectly. Cold wallets are most secure and are completely disconnected from any network. They store the seed on the device itself and also require a password pin to access the device. The advantages to self custody is that you are in complete control of your assets at all times. The disadvantages however is that you are responsible and if you send your crypto to the wrong address there is no refund and if you lose your private key, you lose your crypto.
Joint custody is were you are not the only person that holds the key to your crypto. This can be done in several ways. The most basic is to share the key with someone you trust. This is not advised and rarely done, simply because humans are corruptible. Multi signature or multi sig, is a much more popular method and is done by sharing parts of the key with multiple parties and can be configured my different ways and also gives the ability for smart contracts. Real physical contracts can be made for joint custody in case somebody tries to do anything funny with your money.
Third party crypto custody is were you don't actually hold the crypto kind of like a bank. There are two types of exchanges, ones were you can move the crypto and those were you can't move the crypto and can only trade it in. Either way you technically don't actually own it.
Most favor the ledger cold storage. Well trusted and believed to be most secure by many.