Custodial versus non-custodial wallets
Most centralized crypto services offer custodial wallets, where a third party such as an exchange stores your private keys for you. The exchange might link your wallet to login credentials, but know that if the company is attacked, your assets will be at risk.
Decentralized applications ("dapps"), on the other hand, are non-custodial: you have full ownership, possession and responsibility over your private keys and assets. Wallets come in different types, from browser extensions, to mobile apps, to a USB-like hardware device.
Types of wallets
Crypto wallets can be bucketed into three broad categories:
Web wallets: A self-custody wallet lets you interact with your Ethereum account via a web browser.
Smart-contract wallets: These are wallets that live as a program on the blockchain instead of providing the user with a public and private key pair. These wallets are typically associated with a specific app. Examples include InstaDapp’s DeFi Smart Accounts, Argent, Dharma, Gnosis Safe, etc.