A brain wallet refers to the concept of storing Ethereum in one's own mind by memorizing a mnemonic recovery phrase. If the mnemonic is not recorded anywhere, the Ethereum can be thought of as being held only in the mind of the owner. If a brain wallet is forgotten or the person dies or is permanently incapacitated, the Ethereum is lost forever.
Brain wallets, as apparent from their name, are a type of wallet where the user memorizes the mnemonic recovery phrase of their cryptocurrencies in their brain.
This mnemonic recovery phrase is used to derive the private keys of cryptocurrencies. It’s typically hard to remember or memorize private keys directly because they’re a long string of alphanumeric numbers.
A brain wallet is where a user remembers their mnemonic phrase/private key and never writes it down. And in some cases, if the user forgets the mnemonic phrase or dies without telling anyone or goes into a coma, then the associated Ethereum or crypto-coins are lost forever.