Note: for clarity's sae, this answer will focus on the International Military Tribunals (IMT) and not the various subsequent satellite trials like the National Military Tribunals (NMT) which had a slightly different focus and procedures.

Although the IMT played a little fast and loose with legal norms, the legal foundations justifying the tribunals were laid during the war. The Moscow Declaration of 1943 included a section on atrocities and assured

those German officers and men and members of the Nazi party who have been responsible for or have taken a consenting part in the above atrocities, massacres and executions will be sent back to the countries in which their abominable deeds were done in order that they may be judged and punished according to the laws of these liberated countries and of free governments which will be erected therein.

The Moscow Declaration paved the war for the London Charter of August 1945 which set out the procedures of the IMT. Wartime agreements between the Allies had tabled the idea of prosecution during the war, aside from some minor Soviet trials, and Allied propaganda and declarations did hold that Germans who committed atrocities would be held accountable. These wartime commitments also underscored the need to collect evidence for atrocities and seize relevant government documents as Germany collapsed.

One of the charges leveled against the IMT both then and now was that none of the Tribunal's judges were German, but rather enemies of the Third Reich and thus the Tribunal was a form of victor's justice. However, the Allied powers held that the nature and scope of the crimes were international in nature and would be tried by the Allied powers on behalf of the international community. The international nature and justification for the Tribunal came from several different sources. Firstly, surrender and dissolution of the German government meant there were no Germans or German state left to try its own people. Part of the surrender process included a delegation of authority, including justice, to the Allied Control Council (ACC). Second, even though Germany obviously was not a signatory to policies like the Moscow Declaration, the position of the Tribunal was that such declarations were a sign of a global order that German actions had transgressed. As one of the assistant American prosecutors Lieutenant Commander Whitney R. Harris put it:

the Nuremberg judges declared against aggressive war and related acts which they considered to have been morally condemned by the majority of nations. In the Tribunal's view these acts, like piracy, could no longer be tolerated in a civilized world, and the Tribunal concluded that the responsible individuals could be punished for their actions, just as earlier courts had resolved upon the punishment of men for acts of piracy.

Finally, the Tribunals also claimed that The near universal sponsorship of the UN Charter, as well as subsequent conventions on genocide showed there was a clear endorsement by the international community of the Tribunal system.

The IMT prepared four counts for the accused defendants. Counts II and II were for participation in a common plan or conspiracy for the accomplishment of a crime against peace and planning, initiating and waging wars of aggression and other crimes against peace. War crimes and crimes against humanity constituted Counts III and IV. In hindsight, Counts I and II created a number of headaches for the IMT in that it necessitated the proving of a conspiracy and casting the defendants' actions as part of a coherent plan. But the idea of crimes against peace originated from Justice Jackson's to expand the nature of the trial away from direct actions to a larger indictment of the Nazi system in general. Counts I and II also helped cover, albeit imperfectly, the thorny issue of nullum crimen sine lege (no crime without prior law). The Kellogg-Briand pact and other international agreements created an international standard that either Germany was a party to, or was at least aware of, prior to launching its war of aggression. Conspiracy though still remained hard to prove, and neither German nor French and Soviet law had much of a legal tradition around conspiracy charges, so for most of the IMT, conspiracy remained uncharted legal waters.

Although the excuse "I was only following orders" featured prominently in the NMT and subsequent trials in West Germany in the 1950s and 60s, it was not really salient to the IMT. The defendants were all senior officials of the Third Reich and were the in many cases the surviving executive leadership of the government. Only Wilhelm Keitel and Alfred Jodl pushed the orders defense citing German military traditions of obedience, but the Tribunals' judges rejected this defense. Jodl's judgement summed up this rejection:

His defense, in brief, is the doctrine of "superior orders", prohibited by Article 8 of the Charter as a defense. There is nothing in mitigation. Participation in such crimes as these has never been required of any soldier and he cannot now shield himself behind a mythical requirement of soldierly obedience at all costs as his excuse for commission of these crimes.

The judgement against the military chiefs in the IMT set a precedent for subsequent tribunals that orders could only be a mitigating factor in reducing levels of guilt and the judges of the Tribunals would decide this. In the case of the IMT, because they were service chiefs, they were ones either drafting or directly communicating orders. The orders defense would feature more prominently in the various West German trials of the 1950s and 60s, in no small measure because these German trials were prosecuted under Basic Law, which forbade retroactive justice. Therefore, former SS guards and the like were prosecuted under German criminal law for murder, which allowed for significant leeway in terms of motive and if a defendant could prove some form of coercion, they would receive a lesser sentence as an accessory to murder.

Taken as a whole, the IMT were far from a farce or a kangaroo court. The defendants had access to counsel and a clear outline of the charges against them. Some defendants like Hjalmar Schacht were acquitted completely, and other defendants were able to argue their cases against the counts. Although the IMT did bend a lot of existing international law and precepts, historians' opinions on the IMT have shifted somewhat to a somewhat softer estimation. If this answer were written forty years ago, there would be much greater criticism of both the retroactive justice and the impartiality of the Tribunals' judges. Experience both with genocide trials for Rwanda and the former Yugoslavia have shown the difficulties with bringing war crimes to international trial even after the UN declaration on genocide. Moreover, a good deal of the criticism of the IMT's procedural flaws often stemmed from German jurists' critiques of the system and that existing laws could have sufficed. Historical studies of the German legal system have shown that it was much more Nazified than its postwar defenders claimed and the numerous difficulties German criminal law had with war crimes in the 1960s trials makes such arguments ring hollow in 2016. The IMT was by no means a perfect exemplar of international justice, but given the circumstances, they hewed to legal norms fairly well.