1. GreetingsIt's common to give a little 10° nod of the head and shoulders to greet a friend. A similar gesture can be used to say goodbye.
2. IntroductionsIn both formal and casual introductions it's expected to bow 30° with your upper body. It's important to keep your head and shoulders straight and hands to the side.
After exchanging meishi do a bow and hold for 1 second or so. There's no reason to keep eye contact during a bow (in fact it's considered bad form). Keep a distance to avoid bumping heads (it happens).
If the person you're meeting is very important bow 45°. Never bow and shake hands at the same time.
(Obama was widely criticised for this bow. However, this is a common mistake)
3. Bows of RespectA bow is an expression of humility. It always indicates respect.
Here a caretaker bows to a whale.
In Nara, deer have become so accustomed to bows that they will bow back.
4. Sports BowAnother bow of respect is the bow between opponents before a sports match. This is often a shallow bow of 20°.
5. Religious BowIt's also common to bow to the gods at a Shinto shrine. This is often a shallow bow of the upper body.
Shinto ceremonies often call for a full bow from a kneeling position.
6. Martial Arts BowsJapanese martial arts have their own conventions of bowing. Great respect is paid to your sensei (teacher). It's also important to show respect to your opponent.
7. Bowing to CustomersIn Japan, customers are considered gods (of sorts). It's common for staff to bow to customers. This is usually a bow of the upper body of around 20°.
(employee bowing guide)
(parking attendant bows to a customer in Shibuya)
8. Bows of ThanksIf someone lets you ahead of them in line it's common to give a shallow bow of the head in thanks. It's even common for automobile drivers to bow to each other for small courtesies.
At Japanese weddings it's common for the bride to give an emotional speech to her parents — to thank them for all their support. Here a bride bows as she presents her mother with flowers.
9. Performance BowAs in the West, it's common for performers to bow in response to applause. This is usually a shallow bow. Here Geisha perform a very deep bow.
10. Mild ApologyA mild apology involves a bow with the head of 10°. This can be used if you bump into a stranger or cause a minor inconvenience to someone. For example, if someone holds the elevator doors for you. Say sumimasen (excuse me or I'm sorry).
11. Regular ApologyIf your boss is mad at you — a 45° bow of the upper body is in order. Hold the bow for 5 seconds. Say sumimasen deshita (I'm sorry for what I did).
12. Serious ApologyLet's say you're a company CEO and your company releases a defective product. At the press conference you may apologize with a long 45° bow of the upper body. It may be appropriate to hold the bowing position for 15 or 20 seconds. Say moushiwake gozaimasen deshita (I'm very sorry for what I did).
13. Panic ApologyLet's say you're a waiter and you spill hot coffee all over a customer. You may do a 45° bow over and over again to indicate how sorry you are. Repeat moushiwake gozaimasen (I'm very sorry) with each bow.
This is also how people apologize to the yakuza in movies.
14. Very Serious ApologyLet's say you've committed a serious crime and you're apologizing to the victims. You would bow from a kneeling position. Say makoto ni moushiwake gozaimasen deshita (I sincerely apologize for what I did).
Irrational BowsIn addition to the functional bows listed above there are several irrational forms of bowing. For example, people in Japan sometimes bow when they're on the phone.