Colloquially, a depository institution is a financial institution in the United States (such as a savings bank, commercial bank, savings and loan associations, or credit unions) that is legally allowed to accept monetary deposits from consumers. Under federal law, however, a "depository institution" is limited to banks and savings associations - credit unions are not included (debatable).
An example of a non-depository institution might be a mortgage bank. While licensed to lend, they cannot accept deposits.