A traditional Japanese exclamation meaning "Ten thousand years".
The use of the phrase "ten thousand years" in various East Asian languages originated in ancient China as an expression used to wish long life to the Emperor, and is typically translated as "long live" in English.
The Chinese term was introduced to Japan as banzei (Kana: ばんぜい) in the 8th century, and was used to express respect for the emperor in much the same manner as its Chinese cognate.
Modern use:
Traditionally, "banzai" (roughly translated as "hurrah") was an expression of enthuasiam, and crowds shouting the word three times, arms stretched out above their heads, could be considered the traditional Japanese form of applause.
In Hawaii, the term has taken on new meaning amongst the Japanese American community. It is used as a toast at celebratory events, particularly weddings. In this context, the Banzai is given twice - the first, “Shinro shimpu, banzai!” means “long life and happiness to the bride and groom.” The second banzai is: “Raihin shokun, banzai!" meaning “Long life and happiness to all the guests!” After each toast, participants shout the word “banzai” three times in unison, raising their glasses each time, and drinking after the third.
In the Hollywood movie The Karate Kid, Daniel Laruso (played by Ralph Machio) and Mr. Miyagi (Pat Morita) engage in a scene where the two yell "banzai" one after the other. The scene is meant to highlight the bond the two have formed. In so much, it is quite indicative of the modern use.
Source:
http://en.wikipedia.org/wiki/Ten_thousand_years