The most common sort order
words, names, etc.
It is based on the kana spelling of the word (name, etc.) So if you have kanji in your word, change them to kana. (If you do not understand how kana work, read the writeups on kana and hiragana now.)
If there is a long vowel mark in a word, treat it as the vowel it is pronounced as.
Kill all dakuten and handakuten, and change small kana to large kana. So "joshi" is sorted as "shiyoshi".
Sort these kana strings in the order of the fifty sounds, also known as Japanese alphabetical order. If there is a tie, then names that originally had dakuten come after those without dakuten.
Example: sort these names in a-i-u-e-o order: (1)Daisuke, (2)Jirou, (3)Lina, (4)Ranma, (5)Ryouga, (6)Sakura, (7)Shirou, (8)Tenchi, (9)Washuu.
Kill all dakuten and make small kana big: (1)taisuke, (2)shirou, (3)rina, (4)ranma, (5)riyouga, (6)sakura, (7)shirou, (8)tenchi, (9)washiyuu.
Sort these in the order of the fifty sounds: (6)sakura, (2)(7)shirou, (1)taisuke, (8)tenchi, (4)ranma, (3)rina, (5)riyouga, (9)washiyuu. Seeing as 2 and 7 are tied... we will resolve this later.
Put back the original kana, remembering to break ties by putting names without dakuten first. Result: Sakura, Shirou, Jirou, Daisuke, Tenchi, Ranma, Lina, Ryouga, Washuu.