Banzai is Japanese and means ten thousand years. In the past it was a cheer addressed to the emperor meaning may you live a long life. Now it is a cry of celebration or triumph similar to hurrah or three cheers. It is often yelled after completing a difficult achievement. It is customarily repeated three times with your arms raised in the air. Usage of this has declined in recent years.
The usage of "ten thousand years" originated from China. The phrase was originally used as "cheers to your health" and used by everyone. It later came to be used exclusively to address the emperor as a prayer for his long life and reign. The phrase "ten thousand years" in its original sense refers to ten thousand years of life, and not a period of ten thousand years. The significance of "ten thousand" in this context is only that "ten thousand" in Chinese (and in several other East Asian languages) represents the largest discrete unit in the counting system, in a manner analogous to "thousand" in English.
The term Banzai has had many meanings and usages. It was to praise the emperor, it was used during World War 2 as a battle cry (particularly with kamikaze pilots -this is why the phrase Banzai charge -a final desperate military charge came about), it is the name of a Japanese television show, its the name of a song, it is several fictional characters names, etc.
The picture is from Dr Slump manga by Akira Toriyama, volume 14. It shows the Chinese characters yelling Banzai in joy.