The most important single motive shared by the so-called Axis powers was the establishment of empire. To that end, Germany, Italy, and Japan felt unable to subscribe to the prevailing norms of international law and politics. Each power routinely disregarded international treaties and conventions in pursuit of its aims, often dividing the Allied powers in the process. The Axis powers tested the prevailing system to destruction, and it took some time before the international community was able to respond effectively to the challenge.
The Germans, Italians, and Japanese looked towards a glorious, mythical past that they wished to recover. Virulent nationalism was the dominant political force in each one of these countries, and their governments were obsessed with the idea of restoring national greatness. This would be achieved at the expense of so-called lesser races and peoples: Chinese and Koreans (Japan); Abyssinians (Fascist Italy); Jews and Slavs (Nazi Germany). The international order established after World War I militated against the whole notion of overseas empires created by armed force.
This generated a profound sense of grievance in Germany, Italy, and Japan, who felt that they were being deprived of colonial territory that was rightfully theirs. The Axis powers wanted to achieve great power status, but the Allies wouldn't let them. This fanned the flames of resentment and gave added impetus to nationalist dreams of imperial conquest. The Nazis, for example, were especially obsessed with the concept of lebensraum, or living space. This would involve the forced seizure of large tracts of territory in Eastern Europe to make room for hordes of German settlers.
I think you are referring to the WWII era when all three of these nations made up the Axis Powers. One major thing that they had in common was that the leaders of all three nations felt disrespected by the peace process which ended WWI. Japan's claims in the Pacific were not honored and the Western imperial powers still maintained colonies there. Germany was blamed for WWI and had to pay back billions in reparations to the Allies all while losing major industrial centers. The Entente promised Italy some territory from the Austro-Hungarian Empire if it would join their side in WWI, but this land never materialized.
Another thing that these nations had in common during WWII was that their leadership was taken over by pro-military strongmen who wanted to expand national borders. Hitler went against the disarmament portion of the Versailles Treaty by building up the infantry and air force arms of the German military. Benito Mussolini desired to re-create the old Roman Empire and turn the Mediterranean into a Roman lake once again. Japan sought to make itself the strongest nation in Asia by using its neighbors for raw materials and slave labor. By embracing militarism and expansion, all three of these nations hoped to gain the respect that they felt was not given to them at Versailles.