The UN ICJ can hear two types of cases: legal disputes between states (contentious cases) and requests for advisory opinions on legal questions referred to it by United Nations organs and specialized agencies (advisory proceedings). Only states (UN members and other states that have become parties to the Statute of the Court or have accepted its jurisdiction under certain conditions) can be parties to contentious cases.
As there is no special International Court of Justice Bar, there are no conditions that must be met by state representatives to exercise the right to appear before the UN ICJ, the only exception being that they must be appointed by the government.