The Japanese typically apologize far more frequently than Westerners. This probably results from cultural differences between them. Westerners seem reluctant to admit their own failure. Since apologizing means that admitting one's own failure or guilt, it may not be best thing to do if the problem is to be resolved in a court of law.
Apologizing is considered a virtue in Japan. Apologies show that a person takes responsibility and avoids blaming others. When one apologizes and shows one's remorse, the Japanese are more willing to forgive. There are much less court cases in Japan compared to the States. When apologizing the Japanese often bow. The more you you feel sorry, the more deeply you bow. Click here to learn about bowing.
Here are some expressions used to apologize.
It is probably the most common phrase used to apologize. Some people say it as "Suimasen （すいません）". Since "Sumimasen （すみません）" can be used in several different situations (when requesting something, when thanking someone etc.), listen carefully to what the context is. If you are apologizing that something has been done, "Sumimasen deshita （すみませんでした）" can be used.
Very formal expression. It should be used to superiors. It shows a stronger feeling than "Sumimasen （すみません）". If you are apologizing that something has been done, "Moushiwake arimasen deshita （申し訳ありませんでした）" can be used. Like "Sumimasen （すみません）", "Moushiwake arimasen （申し訳ありません）" is also used to express gratitude.
Formal expression, but it doesn't show as strong a feeling as "Moushiwake arimasen （申し訳ありません）".
Common phrase. Unlike "Sumimasen （すみません）," the usage is limited to apologizing. Since it is less formal and has a childish ring to it, it is not appropriate to use to superiors.
Casual. It is mostly used by men. It also can be used as "Excuse me".
Casual. It also can be used as "Thanks".
Very casual. Adding a sentence ending particle, "Gomen ne （ごめんね）" or "Gomen na (ごめんな, male speech) is also used. It should be only used with close friends or family members.