A trust account is, generally speaking, an account that is held by a trustee on behalf of a beneficiary. Specifically, attorneys hold client funds on behalf of clients. When an attorney earns fees, the attorney, after the client reviews and approves the attorney's bill, transfers client funds from the trust account to the attorney's general account as fee payment of fees.
A trust account in its broadest sense is not necessarily handled by an attorney. Numerous independent trust companies and banks have trust departments that hold assets in trust for beneficiaries. How the accounts work depend greatly on the documents that create the account. For example a trust account that is for a trust containing the term "My wayward son Johnny only has access to this money for issues concerning his health, education, maintenance and support. The trustee has full authory to independently determine the validity of any of Johnny's requests" is not likely to work by giving Johnny a debit card on the trust account.
Rules about how attorney trust accounts work are found in the Rules of Professional Conduct 1.15 a and 1.15b. See the attached links for a full description.