I think that the latter addition of the Good-Evil axis (not sure who decided to introduce it, really) was based on a misunderstanding or at least deviation from this concept and caused more ill than good (imo). Understanding Law quite literally as the kind of stuff lawyers study instead of a metaphysical orientation and attempting to describe fictional characters (that don't belong to the genre in the first place) with this alignment scheme, it was found lacking and expanded upon (I'm theorizing, not saying that's what actually happened). For example "Robin Hood works against the law, but he can't be chaotic, he's clearly a good guy!" or "Batman does illegal stuff but he does to fight the evil guys!". Thus "chaotic good" was born etc. etc.
Not only is trying to apply alignment to Batman a truly foolish enterprise in retrospect, it's also missing the point of what Law means. Batman is, under the old understanding of alignment, through and through a Lawful guy. Not only does he have an incredibly strong moral code he isn't willing to break, he's also unyielding in his quest for justice, order and protection of the innocents. He fights for (hu)man(ity), for civilisation.
My "old" understanding of Law vs Chaos is also informed by ancient myth, where gods made (or shaped) the cosmos (cosmos means orderly in ancient Greek, it's also where "cosmetics" comes from) and man within. The gods wrestled the world from the primordial chaos, as reflected in ancient Greek myths about the fight with the Titans or Mesopotamian accounts of Marduk's fight with Tiamat. Order, or Law, is a fragile island erected by the gods from disorder. Man, civilization and the gods, this status quo, this anthropocentric, inherently human condition. Law already is Good. Batman and commissioner Gordon are immutably on the same side of the grand scheme. Robin Hood is totally a lawful guy, conservative even. He fights to restore the rightful (lawful) king, to restore the old, right way, as decreed by the gods. And Prince John, he's still civilized, he's just a mean dick.
Chaos, that which seeks to upset the human/divine status quo, to ruin civilization, to mock the gods, to change, mutate, ruin and devour is, quite obviously what we humans would call evil. But just like Law is not a simplistic and naive "Good", so Chaos isn't really baddy mcbad "Evil". It is certainly Azaghtoth's mad piping and the untamed bubbling of a primordial mass, the mutation of a beastman, the spell tearing the threads of reality, the uncaring play of the wild hunt, the insane cultist, the inhuman hordes tearing down the city walls.
Alignment means to line up with something, to ally, to stand with someone. Alignment is about where you stand in the big picture. Alignment is whether at the end of the day you fight for the gods, for humanity, for civilization or not. Alignment is not political or judiciary, it's not about individual morals or "goodness or badness". It's about where you're standing in respect to the rest of the universe.
Clerics and divine "magic" is lawful, humans in general are. Elves (if you want demihumans, I actually don't) inasmuch as they are fey and alien, are chaotic (a very non-Tolkenian interpretation). Magic and wizards are, too. (I should link to Lamentations of the Flame Princess here but I can't find the right post.) Etc.
So there you have it, Chaos and Law (although I prefer Order). That's two. Do we need anything else? Well, you could argue that there could be beings and individuals that believe in neither but strive for some sort of balance, like, some crazy dudes in the forest that are all about protecting their village, but aren't above sacrificing a baby from time to time, or madmen actively working to balance the scales, but not favoring either outcome. I'm not a fan of that view but it's conceivable, so sure, let's call it Neutral (or Balance, rather). And finally there are things and creatures that simply don't or can't care. Taking an alignment, allying, lining up is an act of will. Insects, simple natural beasts, automatons, things like that. While it's pretty obvious that a golem commanded by a chaotic wizard is ultimately a pawn of chaos or that one subverted to serve humans is (at least temporarily) on the side of law it's just a mindless tool. There might conceivably also exist people who don't want any of it but just look out for themselves, the world be damned...These are all Unaligned. They have not taken a side, perhaps because they don't want to or maybe they simply can't.
So if you imagine some scales on one side is Order, on another is Chaos. Balance is in the middle, on the tipping point. And Unaligned is off the scales, it hasn't taken any side. And that's how I'm doing alignment.