Admiral Yamamoto was the architect who designed the attack on Pearl Harbor on December 7th, 1941. He had earned an advanced degree from Harvard University during a 15 year stay in the United States before being placed in charge of all Japanese naval forces in 1941. He was actually opposed to the idea put forth by the Japanese army to attack, but once the decision was made he became the mastermind to launch a carrier attack on Pearl Harbor. On that beautiful Hawaiian Sunday morning, Japanese carrier based airplanes struck from clear skies to inflict a devastating surprise blow against the United States Navy forces at Pearl Harbor. The attack was successful beyond their wildest dreams. The "story" goes that when word arrived of the attack's great effect, Yamamoto declined to join in on the frenzied celebrations of his lieutenants. When asked why he was not observing their great success, he quietly replied "I fear all we have done is to awaken a sleeping giant".

Incidentally, "Tora" is Japanese for "tiger", and was designated as the signal for the raid's success. Truthfully, the whole thing is apocryphal -- i.e. made up by Hollywood.
The general consensus is that Yamamoto never actually said either. I personally think "sleeping tiger" is cooler than "sleeping giant" and since they were both probably made up, it's pointless to claim one is more accurate than the other unless you're trying to quote from the movie itself.

The FLYING Tigers, on the other hand, were a group of American volunteer pilots recruited before the US officially entered World War 2 to fight as a surrogate Chinese air force against Japanese forces that had begun their invasion of China in Manchuria in 1937. They would act as a cohesive American unit, but were officially non-affiliated with America because the US was still ostensibly"neutral" at the time. A famous Flying Tiger pilot is 'Pappy' Boyington (Maj. Gregory Boyington), who later founded the so-called Black Sheep squadron that caused such havoc to the Japanese in the South Pacific theatre. By the way, the Americans sighted and shot down the plane carrying Admiral Yamamoto on April 18th, 1943. The plane crashed into a hill in a dense tropical jungle island below. And so it goes...