Despite
Japanese not having one set alphabet, they do have only one set of sounds. Those sounds are ordered. So whether Japanese is written in phonetics, english letters (
romaji), or
kanji, it is possible to
alphabetize it. This is potentially better than the
English system, since japanese alphabetization doesn't depend on
spelling at all. If you can say it, you can put it in order and find it in a dictionary. The disadvantage is that you can't automatically say everything you see (you might not know the reading of a Kanji)
The sound order used in japanese is based on
a ka sa ta na ha ma ya ra wa n
To find the order words are listed, start on the left and sort of matrix multiply that by (a i u e o) in that order.
So the order things are listed in is
a i u e o ka ki ku ke ko sa si su se so etc.
So, anything starting with ka is before anything with sa, and ka is before ki. Also, ka/ga, sa/za, ta/da, and ha/ba are considered the same, and mixed in. Just ignore the little tenten or the dot.
This alphabetization system is used everywhere we would use the alphabet order, so it's common: dictionary, video stores, libraries, lists of names, etc. Most Japanese schoolchildren know it, but it's not mentioned often. If you don't know it, you can bumble around for quite a long time.
One peculiarity of this system is that kanji are ordered by their pronunciation in that context. So if you know one pronunciation of it, but the current one is different, you're stuck. This is a problem because many kanji have more than one pronunciation. Fluent speakers won't have a problem with this, but someone trying to figure out with a dictionary might have some trouble. Names, however, can trouble even Japanese people, since there can be 20 pronunciations for one name kanji, and which one is correct is totally arbitrary.
This system may seem strange, because "letters" in Japanese are divided differently than we do. A letter can be a single vowel, or a combination of a consonant and a vowel, or a solitary N.
a ka sa ta na ha ma ya ra wa n is also the first line in the Japanese version of the alphabet song:
akasata-na hamayarawan
ikishichi-ni himiri
ukusutsunu fumuyuru
ekese-te-ne hemere
okosotono homoyo-ro
It doesn't have much meaning, but it can be broken up to sound like japanese words. The particles indicated with hyphens have meaning: ~na is "~esque", ~ni is "in ~", ~te is like verb+ing, ~ne is "isn't that right? ("eh" if you're canadian), and ro is a more colloquial form of ne.
I was taught this by a 13 year old Japanese student of mine during cleaning time in a Japanese school.