According to both the novel and the series, the main split between the two timelines began with the successful assassination of President Franklin D. Roosevelt in 1933 by Giuseppe Zangarast. Afterward, the United States is subsequently led by John Garner (FDR's vice president) and then by Republican John Bricker. These presidents led to the destabilization of the country and both failed to enact the proper reforms to pull the country from the Great Depression. They continue to maintain an isolationist philosophy during the war and, as a result, they are unable to support the United Kingdom and the Soviet Union's fight against the Greater Nazi Reich or defend themselves from the Japanese Empire in the Pacific.
In the following years, Nazi Germany conquered the Soviet Union and exterminated most of its Slavic peoples, and either invaded or forced the surrender of the United Kingdom. The Japanese Empire then destroyed most of the United States Navy fleet in a decisive battle at Pearl Harbor (much more destructive than in our timeline), and conquered in short order Hawaii, Australia, New Zealand and the rest of the Oceania. By 1945, the Axis Powers were strong enough to launch a successful invasion of the US from both the Pacific and Atlantic coasts.
The Nazi's then detonated the first hydrogen bomb (dubbed the Heisenberg Device) on Washington D.C., forcing the United States to surrender by late 1945. Despite this, the Axis Powers were forced to fight a hard and tenacious American Resistance for almost two years until September 18th, 1947, when the Nazi Reich declared that the resistance has finally been put down. This day was marked as Victory in America Day (or VA Day) which officially ended the conflict with an Axis victory.
Sir Winston Leonard Spencer-Churchill was a British statesman and the Prime Minister of the United Kingdom during the war. He was also an officer in the British Army, a non-academic historian, a writer, and an artist. His fate is unknown, although most likely in exile or executed.
Joseph Vissarionovich Stalin was the leader of the Soviet Union from the mid-1920s until his death in 1949. When the Nazis won the war, Stalin was executed, his death effectively caused the collapse of the Union.
Franklin D. Roosevelt
Franklin Delano Roosevelt never managed to become President of the United States. He was assassinated in 1933 by Giuseppe Zangara, an Italian anarchist who apparently expressed a "hate for all rulers". As a result, the USA never recovered from the Great Depression and was incapable and unwilling to help the Allies during World War II, which led to a German victory in Europe and Africa, and a Japanese victory in Asia and the Pacific.
The series offers little information about Roosevelt, although it does mention he was assassinated before taking office. While the TV series doesn't specify the identity of the murderer or when the assassination took place, it's generally assumed to be the same as in the novel. The characters in the series occasionally discuss Roosevelt's demise. They privately wonder if the world would be a different place if he had survived his murder.