I feel like the title says it pretty well. Alphabetical order is commonly used for organizational systems in places which use the Latin alphabet for writing, but it seems as though it's just an arbitrary list everybody memorizes rather than following some system like putting the vowels first or listing the letters by the frequency of their use.
How did alphabetical order get decided? Were there different countries that had different orders who had to reconcile those systems?
Also, what's the story of alphabetical order for other alphabets like Cyrillic and Arabic?
Arabic
Arabic has two alphabetical orders. Arabic is, like almost all semitic languages, an "abjad" (or a pseudo-abjad, for reasons that aren't really relevant here). The name comes from the first letters of the Arabic alphabet in this ordering "Alef" (A), "Ba" (B), "Jeem/Geem" (J/G), "Daal" (D). This ordering is basically derived from the ordering of the older Aramaic alphabet from which Arabic lettering is derived.
The full ordering also allows you to treat letters as numbers, which can be seen here: https://en.wikipedia.org/wiki/Abjad_numerals
The second, more modern order arranges Arabic letters partly by shape, so that, for example, the letters "Ba" "Ta" and "Tha" which only differ in the place and number of their dots, are placed next to each other.
This is now the most commonly used alphabetical ordering in Arabic in, say, dictionaries or to teach children, but you will occasionally see, for example, series of arguments or points laid out in the old system in the way that in English you would write Clause A, Clause B, Clause C, etc.