A "banzai run" or "banzai charge" (バンザイ突撃) is the name given to a human wave-style attack by the Allied forces in World War II. The charge is when mounted infantry forces of the Imperial Japanese Army. The name Gyokusai (玉砕) was also applied to the charge.
The charge is usually a suicidal attack launched to avoid capture, surrender, or perceived dishonor in a final attempt at maximizing the odds of winning in the face of usually superior forces, as was the case in WWII.
The Allied troops applied the name "banzai charges" to these kind of attacks due to the Japanese Army's practice of shouting, "Tennōheika banzai!" (天皇陛下万歳!) as they execute the attack. It means, "May the Emperor reign for ten thousand years!"
It's important to note that though Westerners used the world "banzai" to describe the attack, the Japanese never used the word this way. The word "banzai" (万歳) literally means "ten thousand years" in Japanese. It is a common expression that is used to celebrate long life in Japan, essentially wishing for something or someone to live for eternity in good health.