A “brain wallet” is the practice of memorizing a seed phrase that is converted into a bitcoin private key. They present a major risk of theft and loss.
One major problem with brain wallets is a lack of true randomness. Attackers can guess and check many brain wallet seeds quickly in parallel. As of 2016 researchers reported an active community of thieves checking brain wallets for balances and draining them within minutes of creation.
Brain wallets also present a significant risk of loss. Injuries or just forgetfulness can cause a seed to be lost forever. To be of comparable security as a normal bitcoin key, a brain wallet seed phrase has to contain a large number of words, such as memorizing a 12-word BIP39 seed phrase. If any of the words are missing or remembered out of order, it may be hard or impossible to recover.