The agency problem strikes me as an ethical, practical, and economic issue in equal measures.
The agency problem is one in which one person or entity acts on behalf of another. This might apply to managers or executives of corporations acting on behalf of shareholders, a lawyer representing a client, or a stock broker or hedge fund investing on behalf of a client. In each case, the agent interacts with some person or entity on behalf of a third party. The problem lies in the degree to which an agent acts out of self interest or to benefit the principal.
For example, an executive personally benefits from having a multi-million dollar salary, luxurious office, private plane, and stock options. One may question, however, whether stockholders or customers benefit from that. Lawyers in a divorce case make more money the longer the lawsuit is drawn out and the nastier it becomes, but that certainly does not benefit the divorcing couple or their kids. A stock broker who frequently "churns" a portfolio may make more money on fees but not better returns for a client.
In all of these cases we have several issues. The first is the ethical one of fiduciary duty. This involves an element of trust; when it breaks down, this undermines much of the basis of advanced industrial economies. Social attitudes such as trust are correlated with economic development; if you cannot trust a bank, for example, you will not save and invest but might keep money under a mattress. Thus the ethical issues become economic ones.
On a practical level, there is the question of how one can incentivize agents to behave in a manner that benefits their principals, and how to align transactions such that the self interest of the agent coincides with that of the principal.
The agency problem refers to the dilemma whereby a conflict of interest arises whereby a party is expected to act in the best interest of another party. Since we are motivated by self-interest, conflicts may arise as a result of the representative party acting in a manner contrary to the best interest of the patron.
The agency problem is primarily and ethical issue (although it does have economical elements) as contracts usually stipulate for explicitly and implicitly that an agent should act in the best interest of the principal. A violation of this is an ethical infraction, although in many cases it can also lead to prosecutions or loss of professional licenses in regulated industries.