A basic question for students of Japanese and people new to Japanese culture is when it is appropriate to bow. The answer is almost all the goddamn time. You bow when you say hello, when you say goodbye, when you're thanking someone, when you're apologizing, when you're confirming instructions, when you're embarrassed, when you're grateful. You bow while you're driving your car. If another car lets you change lanes, you keep your hands on the wheel and bow your head towards them. You even bow while talking on the phone, but that's more reflexive than an actual requirement. When you're meeting someone important, you bow several times throughout the conversation, and when you leave, you back out of the room bowing.
Situations where it's not appropriate to bow: When insulting someone, when talking to your children. There are probably others. Bowing obviously becomes less frequent the more familiar you are with a person. Then there's the question of how to bow. Hands at your side, bending from the waist, the deeper the bow indicates greater respect. It's important to give a good, solid bow when you're meeting someone, but conversationally, a bow can be little more than a nod.
A situation when it is unambiguous as to whether you should give a nice, deep bow: when you're a foreign leader meeting the emperor of Japan.
The objection to Obama's bowing comes off as totally absurd to me, but I thought I'd address the right-wing criticisms anyway.
US Presidents shouldn't bow to foreign leaders
It's called cultural sensitivity, and it gets you a long way in diplomatic relations. Anyone who's been to a foreign country knows how much more respect you get by just learning how to say "Hello" and "Thank you" and the very basic customs and taboos than by walking around in your "USA is #1, bitches" T-shirt and loudly expecting everyone to accommodate you. Bowing when you meet someone is kind of the bare minimum you can do in Japan, and if US Presidents really never bow to Japanese leaders (as the right claims) then no wonder we come off as arrogant pricks abroad. If we were a Muslim country, that might be different, because the Koran forbids bowing to other people*, but as far as I can tell there's no big ideological conflict in bowing for most Americans. Just a common nicety to show that we respect their culture.
*Which makes me think that it couldn't have been a cultural requirement to bow to the Saudi King. Hmmmm.
The depth of Obama's bow indicates weakness to the Japanese
Now this argument is a bit more sophisticated than the ol' "cultural sensitivity is for pussies" one. True, Obama's deep bow indicated great respect, and between Japanese people hierarchical imbalance does play out with the relative depth and shallowness of bows. However, the general view that Japanese people have is that their culture is so intricate and complicated that no one outside of themselves could possibly understand it. Thus, when a foreigner uses Japanese words, handles chopsticks, or does something sociolinguistically appropriate like bowing, no matter how sloppily said foreigner executes these things, they are usually met with some delighted, awestruck Japanese people. Sometimes they even start clapping.
Obama's bowing form was a little off. He shook hands and bowed at the same time, while you're supposed to do one or the other (though I've seen many Japanese people shaking hands with foreigners and bowing at the same time because it's such a reflex). However, he's being judged on the foreigner scale, rather than the Japanese person scale, which means that they'll read little more into his bow than "Oh my god, he made an effort! He is such a cool guy!"
I'm sure the Japanese reaction to Obama's bow was that of being incredibly impressed and flattered. For some reason, it's the domestic reaction that's a problem. Their outrage is masked under the guise of concern for foreign policy, but if they stepped out of their ethnocentric bubble for a minute and bothered to learn about foreign customs, they'd discover this outrage is unwarranted. I suspect many of these pundits know their outrage is unwarranted, and they just want to rile people up. Really, it's not the foreign policy repercussions they find offensive but cultural sensitivity itself. When Americans are offended by showing graciousness to a foreign leader who is our ally, who consistently supports us and honestly thinks that America is the shit, what do they suggest we do to foreign leaders with whom we are on tenuous terms? I think I know the answer. And I think it's no kind of future for productive foreign diplomacy.
Edit: Props to Harry for encouraging me to write about this, because I totally wasn't going to.