This stuff depends heavily on the medium in consideration. Usually arcane magic is basically a science made magic, as if there is some kind of stuff lying around and using it; divine magic is some kind of magic channeled by some will of a god or by some kind of ideal or will ( there are feats of godless divine magic, thought that can sound weird, examples of this are some clerics in ded next and in world of warcraft ); then there is a bunch of various subclasses of magic that are "elementalish" or "naturalish" that are somehow similar to arcane magic in the sense that develop something that is in the world but not by studying it rigorously but more at will, and usually this stuff mixes with arcane magic because it acts the same; and then there is mental magic or psionics where magic is some kind of force or feat developed by a special brain training that allows the user to shape stuff.
As you can see, it depends a lot on interpretation, but also many things are just very similar but with a different name or focus to have flavour.
Giving a rough approximation, pretty much every "mage-ish" user in lol is "arcane", from xerath to galio. They both use energy stored around ( or inside ) them to perform stuff, some do it in a more complex fashion ( like Ryze ) and others just unleash it ( like Lux, Xerath or Taliyah ) with different levels or complexity, or others just use magic without using proper spells ( like Poppy). Even Aurelion Sol feels a lot arcane to me, just in a very innate way and with a stellar flavour.
The only proper divine users i can think of, just as a quick thought, are Illaoi, Bard, Leona, Diana, Taric ( see how these last three refer to a deity-like patron?), Azir, Soraka and maybe Karthus ( yes, deities can fancy undead too).
Obviously that may depend a lot on how the LoL universe is structured, this is just an approximation.
