OK, so what exactly is all this bowing stuff about?
In the Japanese culture bowing is probably a rough equivalent of shaking hands or giving a "thumbs up". It is a form of acknowledgement, a way of greeting someone and also a sign of respect.
Part of Karate etiquette is bowing at certain times.
There are three types of bow, the "small" bow, the "full" bow and the "kneeling" bow.
The Small Bow
This is used as a form of acknowledgement and a sign of respect. In practice, it is a small nod of head, usually keeping your eyes ahead or on your opponent. Whenever you enter or leave the Dojo (training hall) you should pause a second and make a small bow to show respect for the place where you learn to grow as a person.
When facing a partner or opponent, before you start and after you finish, you give them a nod of the head as a way of saying "hello", "goodbye" or "thank you". It is also used to show that you are putting any personal feelings to one side and that while you are training, sparring or fighting you are acting purely as a Karateka.
The small bow is also often used as a "yes" or "thank you". The instructor or your partner may correct a mistake you have been making and you perform a small nod of thanks. When asked if you understand what has just been said or demonstrated a small bow often accompanies "oos" or "hai" (hai / "yes" is technically more correct than oos which is a very shortened version of ohio gosai masu / "good morning").
The Full Bow
A much more formal type of bow, the full bow is slower and lower.
The formal bow is mainly used at the start/end of kata is a fairly rigid bend from the waist. As you have no opponent other than yourself there is no need to keep your eyes forward.
The Kneeling Bow
The "seiza" or formal kneeling bow at the start and end of a session is the most misunderstood and follows a strict ritual. The bow is performed from a kneeling position and is a very deep bow bringing your forehead almost to the floor. The origin lies in the Samurai culture, where the Samurai warrior had the power of life and death over the rest of the population. If you offended a Samurai they were within their rights to whip out their sword and chop your head off. The full bow is a sign of deep respect and trust, offering the back of your neck to the other person's sword.
When all are lined up for the seiza ritual, it is ONLY the most senior graded person who is entitled to stand at the front. Sensei Whale is currently the highest grade in Sandokai, so he is the only one who will automatically stand "out front". If there were another person of the same grade they would both take their position at the front of the class. The senior grade then has the option to invite other senior grades to join him/her at the front.
In a club setting where there are only two or three black belts present it may be that all Dan grades are invited. At a course or grading where there are a significant number of black belts attending it may be only the Sensei (3rd Dan and above) that are invited leaving the 1st and 2nd Dan black belts lined up with the rest of the Kyu grades (students). This has the practical purpose of avoiding a situation where you have 20+ black belts trying to cram themselves into one line! As a rule of thumb, it is usually the top two or three highest ranks out front.
The senior grade will then give a nod to the highest grade facing the instructors who will give the command "Seiza". All will then adopt the kneeling position. The senior grade will kneel facing the students, and the other instructors with him/her will swivel about 45 degrees towards the senior grade so they bow to both the senior grade and the students at the same time. If it is only Sensei facing the students, the command is then Sensei ni rei (bow to teacher/s), followed by the bow and then the other seniors will turn to face the students. If there are 1st and 2nd Dan Sempai at the front then after the Sensei ni rei, after the other seniors have turned to face the students, there will be a second bow - Sempai ni rei (bow to seniors). As this is a bow for the Sempai, any Sensei present will not bow (they've had their go!).
If there are no Sensei present, then only the Sempai ni rei command is given. In the unlikely but possible situation of no black belts being present, blue and brown belts are also considered Sempai, and the highest graded Sempai will take the Sempai ni rei bow, either standing or as a full seiza. If a class is taken by anyone lower than blue belt (it has happened - the first karate club in Cornwall was started by a green belt) then a standing small bow with a simple "Rei" command would be appropriate.
After the bow (or bows), the instructors at the front will stand in grade order, the highest grade first. Then the kiritsu (stand) command is given and the students stand as a group. A small bow follows, and the senior grade will then turn and do a small bow to the other instructors who then rejoin the students (unless instructed otherwise).
There is one other small point of etiquette that takes place only at the end of a session. The command Sensei ni rei is changed to Sensei ni, domo arigato gosai masu, rei. Domo arigato gosai masu simply means "thank you", and saying "thank you" at the start of a session is a bit pointless, as you don't have anything to thank the Sensei for! It is only the Sensei that receives the "thank you". The bow to the Sempai remains unchanged.