The Allied Powers
The Allied powers had less in common with each other than the Axis powers (which were united in their fascistic and expansionist ideology), but they had one central mission: to stop German, Italian, and Japanese aggression.
The largest three Allied powers were the United States, United Kingdom, and the Soviet Union, with affiliate states including the Republic of China, France, Poland, Canada, Australia, New Zealand, South Africa, Brazil, Mexico, and many others. The Allied powers would later go on to be the first countries admitted membership in the United Nations.
The United States
The United States had hardly fought in World War I, joining only a year before the war ended, on neither side. By the time World War II started, the US was faced with an economic disaster: the Great Depression. In 1929, the stock market crashed and resulted in a quarter of people losing their jobs. The gross national product was cut in half, industrial production was halved, construction of factories and plants fell 90%, automobile production dropped by two-thirds, and most existing factories and plants operated at a fraction of their capacity. The stock market's value declined 80% overnight, wiping out many people's assets and retirement funds (source: Shmoop).
Rather than turning to fascism and radically authoritarian ideology, the United States experienced a leftist populist wave, culminating in the election of President Franklin D. Roosevelt in 1933. He would not leave office until he died, in 1945.
America would not enter the war until after the attack on Pearl Harbor, on December 8, 1941. The US would cooperate with the Allies in the invasion of Normandy, and would be the main force combating Japanese aggression in the Pacific theater, ending the war with Japan with the atomic bombings of Hiroshima and Nagasaki.
Great Britain
Like the United States, Great Britain suffered the effects of the Great Depression. By 1930, the UK had lost over 5% of its GDP. The UK's Prime Minister, Neville Chamberlain of the Conservative Party, was best known for his appeasement foreign policy and signing of the Munich Agreement, which gave Hitler the Sudetenland.
With Hitler's invasion of Poland; however, Chamberlain declared war and led the war effort against Germany for eight months, before resigning and being replaced with Winston Churchill. The UK was the main adversary for the Germans until the Soviets and Americans entered the war, following the Nazi invasion of Russia and the attacks on Pearl Harbor respectively.
Communism
The ideological division between the Soviet Union and the other Allied powers would become the focus of the Cold War. The Soviet Union was a communist state: there was increased emphasis on an individual's role within a collective, rather than on the individual's rights. Communism sought to overthrow the bourgeoisie and seize the means of production, ending worker exploitation. Stalin's flavor of communism was more nationalist, seeking Socialism in One Country rather than global, permanent revolution as envisioned by Leon Trotsky. Stalinism centralized the state and made Stalin a cult-like figurehead.
The Soviet Union
The Soviet Union's involvement in World War II wasn't always on the side of the Allies. In fact, in 1939, Nazi Germany and the Soviet Union signed a non-aggression agreement called the Molotov-Ribbentrop Pact, which guaranteed that neither nation would ally itself to or aid an enemy of either state. This agreement would be used to allow Joseph Stalin to invade Poland, along with Hitler.
The non-belligerence pact would become null and void following Operation Barbarossa, the Axis invasion of the Soviet Union, on June 22, 1941. Hitler invaded the Soviets for purely racialist reasons, regarding Slavs and Russians as sub-human Untermenschen.
The Soviet Union's initial friendship with Nazi Germany made the Allied powers uneasy following the end of World War II. This uneasiness in ideology and trust became the spark for the Cold War.