To own or control cryptocurrencies and assets inside blockchains they are all under public keys, also known as accounts or addresses. Each of those accounts are controlled by private keys. A private key is like a unique password that controls a public key. When someone has a private key, they have absolute control of an account that is associated to that private key in a blockchain. Nobody else has such control.
Custodial
However, some people don't have the private keys to their cryptocurrencies. Instead, they delegate to another person or service the "custody" of their assets, so the other person or service has control of the private keys to the accounts. This means that you are trusting a third party with your assets.
When you have delegated the control of your assets to a third party, this kind of relationship is called "custodial" because the service, which may be an exchange or a wallet service, really has ownership and control of the crypto. You just interact with them through their app or website. This is very similar to how banks or brokers work today.
Non-custodial or Self-custody
With non-custodial or self-custody relationships or services such as Emerald Wallet, you have total control of your private keys, and therefore of your accounts inside the blockchains and your assets. This is a lot of responsibility, but also much more secure and the purpose of the blockchain industry!
The way Emerald gives you this control and security is by having only options to create wallets inside Emerald that use hardware wallets, seed phrases (24 word secret phrase), your raw private keys, or Private Key JSON files.
Emerald does not, and will never have your private keys.