Ichidan verbs ditch ru, godan verbs replace the next character with any of the 5 vowels. Is it known how this distinction came to be? Were the irregular verbs kuru and suru always irregular?
The theory is this:
Godan verbs have a consonant at the end of their root. The various stem forms each add a vowel on the end, and then further suffixes are added. This has been the case in Japonic as long as we can tell; though /o/ is a new vowel to the paradigm (they used to be called yodan verbs).
Ichidan verbs with -iru (kami-ichidan verbs) probably had a vowel at the end of their root, initially. I'm not sure why the shuushikei added a -ru; this might have been on analogy with the rentaikei (which uses -ru in several other verb classes, and later replaced the shuushikei forms in most cases). I don't think anyone's ever explored the question of why the root-final vowel is always -i, though.
It's not fully clear whether or not all the vowels godan verbs take could have at one point been added to ichidan forms. Old Japanese is very aggressive about deleting vowels adjacent to other vowels, though, and so it wouldn't be surprising if they had been there and were just all deleted. (I don't think Ryuukyuuan languages show any trace of these vowels either, though.)
Ichidan verbs with -eru (shimo-ichidan verbs) were all once nidan verbs, which had two vowels in the stem: -e- in most forms, but a shuushikei in -u and a rentaikei in -uru (so modern 食べる was once 食ぶ in the shuushikei, and 食ぶる in the rentaikei). These are mostly believed to come from the incorporation of 得る 'get', which before the creation of nidan verbs would have just been irregular. These have since lost the alternations involving an -u, and have become shimo-ichidan verbs.
(The only historical shimo-ichidan verb was 蹴る, which is now a godan verb - go figure.)
する and くる seem to have been irregular as long as we can tell. People have tried to propose various unified underlying root forms, but none have been all that plausible. They're a bit less regular in Modern Japanese than they used to be, but even so, the vowel alternations they underwent in Old Japanese aren't the same as the vowel alternations for yodan verbs.
Incidentally, する and くる aren't the only irregular verbs in Modern Japanese. 行く has the unexpected forms 行った and 行って instead of *行いた and *行いて, and 良い has an irregular shuushikei/rentaikei いい instead of the expected (and historical) よい.