The agency problem is covered in full in the entry Principal-Agent Problem, which explains the theory, classic examples and the ways firms try to fix it. This page gives a short definition and explains the different names.
The agency problem is the conflict that arises when one party, the agent, is hired to act for another, the principal, but has its own goals and knows more than the principal can easily check. Company managers acting for shareholders are the standard example: they may favour empire-building or a quiet life over profits. The gap in knowledge is a form of asymmetric information, and the agent's freedom to take risks someone else pays for links it to moral hazard.
Economists use "agency problem", "agency dilemma" and "principal-agent problem" for the same idea. The money spent monitoring agents, plus the value lost when they still misbehave, is measured as agency costs. Remedies include bonuses tied to results, oversight by the board of directors, and pressure from shareholders.