The Axis Powers in World War II mainly consisted of Germany, Italy, Japan, and smaller countries that sided with them (such as Hungary and Romania). The Axis Powers main goal in WWII was to, first and foremost, destroy their enemy, the Allied Powers. The Allied Powers consisted of the United States, Great Britain, France, China, and other countries such as Greece, Mexico, India, and Poland. Needless to say, the Allied Powers has quite a few more big powerhouse countries within it than the Axis Powers did.
The Axis Powers main role in the second World War was to abolish communism in the Soviet Union, expand their territories, and the end the restrictions of the Treaty of Versailles that was formed at the end of World War I.
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The Axis Powers were the coalition of countries that fought against the Allied Powers, and were headed by Germany, Italy and Japan. The major role of instigator can be said to have been played by Germany, with the other countries becoming a part of the war by forming alliances and treaties. The three major countries came together as a result of the Anti-Comintern Pact, aimed at defending against communism.
The aggression of countries like Japan and Germany in invading it neighbours and the simultaneous fall of the League of Nations toppled the world into war. The objectives of both were similar in creating a united state of their 'own' whether it be 'Aryans' for Germany or 'Asians' for Japan.
For a more in depth look at the exact events that lead to the war refer to the link. It is an excellent study guide that concisely explains the details.