Customary international law is one component of international law. Customary international law refers to international obligations arising from established international practices, as opposed to obligations arising from formal written conventions and treaties. Customary international law results from a general and consistent practice of states that they follow from a sense of legal obligation. Two examples of customary international laws are the doctrine of non-refoulement and the granting of immunity for visiting heads of state.
The International Court of Justice (ICJ) is the main judicial body of the United Nations, and it settles disagreements between member states of the United Nations. Under Chapter II, Article 38 of the Statute of the International Court of Justice, international customs and general practices of nations shall be one of the court's sources of customary international law is one of the sources of international law. Customary international law can be established by showing (1) state practice and (2) opinio juris.
For more on customary international law, see this Harvard Law Review article, this University of Virginia Law Review article, and this University of Michigan Journal of International Law article.
Malcolm N. Shaw, International Law 80 (5th ed., Cambridge, 2003).