Gambari is the Japanese national spirit of patience and determination. After World War II Japanese reconstructed their ruined country to become a powerful economic nation. Japanese are often called workaholics. Behind this non complaining attitude of the Japanese, is gambari spirit.
Word gambaru is frequently used in Japan. It literally means "Do your best! Hang on there! Keep going!". It's used in workplace and in any situation that would require person to give his/her best efforts. Typical example would be a school exam. Often when saying goodbye, Japanese also say "oshigoto gambatte" which means "do your best at work". Athletes also gambaru (work hard) to win their games and get medals.
Some Japanese have recently awakened to realize that one shouldn't use this word when person is suffering from depression or has too difficult life situation, as it quietly suggests that one should act strong even in circumstances when it's a sheer impossibility. Japanese is known to be only language in the world that has a world of death from overwork, karoshi. This is typically a heart attack but can be also a failure of other parts of body, or a stroke. Karoshi is on high increase in Japan. Number of claims filed by families for compensation as a result of death from overwork is about 500 a year. However, the real figure is said to be several thousand.
There are changes in gambari spirit in Japan and concept of free time is now getting more popular. Recently, for example five-day workweek has became more popular. Ministry of Education decided to reduce the content of study in elementary schools in their programme started 2002 "Education in relaxation"). The word might be little too optimistic though.
Foreigners usually tend to feel more negative about gambaru than Japanese. The meaning of the word however isn't "if you fail to do your best, you won't be accepted". Maybe it's rather "do your best in your own way".