In law, fiduciary refers to something that involves trust, in the relationship between the trust and the trustee. In finance, the term refers to something whose value depends on its securities or the issuer's reputation. For example, paper money used to have a value in gold, but now the value rests on the institution that prints the money.
Fiduciary can refer to a person or a legal entity - a bank or a financial firm. They have the responsibility and power to act on behalf of one another. This is especially valuable in situations requiring total trust and honesty; for example, investors have advisors bound by fiduciary rules to look for their clients' best interests.