This is a little confusing for someone who didn't grow up with the Japanese language. You have fifty sounds. Then, they're in a certain order. So, these fifty sounds are the Japanese syllabary. That's hiragana / katakana, probably from back in the day when there were more sounds than there are now. So, the "fifty sounds in order" makes up aiueo order, where the "aiueo" is あいうえお representing the five columns.
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.
The town names are listed in aiueo order.
In this list, the students’ names are placed in aiueo order by surname.
The first Japanese dictionary arranged in Japanese syllabary order was the "Wakun no shiori."