"Shazai (謝罪)" -- from "The Japanese Tradition" series of videos by Japan Culture Lab -- is a useful and entertaining guide to the art of apologetic bowing.
+ Video
The video provides techniques and tips for the entire spectrum of bows, from the commonplace shallow bow (for casual apologies) to the various forms of ojigi (for serious apologies) -- including the long ojigi (used when apologizing to the public for a scandal or product recall) and the perpetual ojigi (to express determination). Also explained is the kneeling bow (used predominantly by ninjas), the grovelling dogeza bow (used when you are unequivocally in the wrong, such as when "caught red-handed in an orgy of evil"), the momentous doge-fuse bow (for the ultimate apology), and the doge-umari bow (the final straw).