That's what I found as a response to your question:
Basically, International Law is divided into Public International Law (law of treaties, conventions, bilateral or multilateral agreements, all of which are negotiated between sovereign states by mutual agreement - for instance border issues, human rights issues, environment, security, laws of war, intergovernmental institutions, etc.), and Private International Law (commercial law governing trade between countries, rules of international trade, intellectual property, etc).
My current work is in international criminal law, which is mostly treaty based, but is also a patchwork of international courts and tribunals (ICTY, ICTR, ICC), state practice, and some evolving principles of universal jurisdiction - i.e. put simply, the principle that some crimes, like genocide, are so grave, that they may be tried in certain jurisdictions regardless of whether the perpetrator or victims were from that jurisdiction.
About employment...Some firms indeed serve as consultants to governments and international institutions on public international law, but the vast majority of these advisors tend to be NGO experts, academics or government advisors. Most big New York firms with an "international law" practice will typically work on things like acquisitions and mergers, investment banking, etc. which require knowledge of domestic laws in individual countries. Often a firm will have offices in different countries with national legal experts who work transnationally as a team.
Some of the most exciting opportunities in international law these days are in public international law. There have been historic developments in the last 10 years which have led to the enforcement of human rights and humanitarian law (laws of war) in international tribunals. You could work for international courts or tribunals, human rights organizations, think tanks, governments, the UN, universities, etc. The quality of life is much better than in large firm (it can be very tiring with 80 hour work weeks...yikes!) and you would be making meaningful difference in the world...