Sun Tzu, also known as Sunzi, was a Chinese military strategist, general and philosopher who was thought to have lived from 544 BC to 496 BC. Sun Tzu is the author of The Art of War, a work of military strategy that has had enormous influence in Western and East Asian philosophy, politics, the study of International Relations and on business decision-making processes.
Sun Tzu's birth name was Sun Wu, but he is known with the honorific title "Tzu" in Western Society, which means "master", so his name translates to "Master Sun".
Sun Tzu is also known as Sunzi, but they are romanizations of the same name, using difference systems. Sunzi is rendered using the Pinyin system and is closer to the correct pronunciation of Sun Tzu. Sun Tzu is often mispronounced as Suhn Tsoo due to the attempt to phonetically read this specific transliteration (rendered with the Wade-Giles system).
The right way to pronounce Sun Tzu is actually closer to SOON-tzuh. The "u" in Sun is pronounced with a double "o" sound. The "tz" sounds like the "zz" in "pizza" and is followed by what sounds midway between an "eh" and an "uh" sound, as heard in the audio pronunciation. This last sound is quite short and is essentially the sound you make when you try to pronounce the "zz" sound in "pizza" on its own.