I like the old alignment grid, but personally would be fine with a slight modification. Namely, I'd add "unaligned", which would become the default alignment for more or less everyone. Alignments themselves would become more extreme, and involve either innately being born into that alignment, for some magical beings, or a magical ritual that dedicates you to the alignment you match up with, for people.
In this system, Good people are saintly, and wish to use their strength to help others however possible. They willingly save kittens from trees, villages from bandits, and starving pickpockets from a life of crime. The end mission for "Team Good" is to make the world the best place it can possibly be. The archetypical "knight in shining armor" is Good.
Lawful people are regimented and orderly, and want the entire world to be the same way. Dissent, disorder, people living outside the bounds of society, and confusion are anathema to them. The end mission for "Team Law" is to bring the entire world under strict, totalitarian rulership. The Borg are Lawful.
Chaotic people are free spirits, who cannot abide the yoke of civilization, the subjugation of the strong to serve the weak, and the century-spanning and immutable hierarchies that define many societies. They much prefer self-reliance and respect what a strong person can get for himself. The end mission for "Team Chaos" is to dissolve civilization, bringing the entire world into anarchy or tribalism. The archetypical orcish barbarian is Chaotic.
Evil people are real jerks, who use their power to actively harm others whenever feasible. If they do not believe it will bring swift retribution, they will lie, cheat, steal, plunder, and murder to get ahead, or just for the sake of hurting other people. The end mission for "Team Evil" is to make the world a really rotten place, where everyone suffers all the time. Or is dead. The archetypical "lord of darkness" is Evil.
Neutral people are weirdos and Rilmani, who for some reason believe very strongly in opposing all of these forces. While some opposition to Law and Chaos and strong opposition to Evil is natural for most Unaligned people, trying to intentionally bring some sort of weird balance to the universe isn't. "Team Neutral" doesn't really have an end mission, they just want to keep all the aligned forces in balance eternally. As such, the Neutral alignment is the rarest of all of them, usually only found in rare groups of Druids, eccentrics like Mordenkainen, and strange magical creatures like Rilmani and Concordant Killers.
The "double" alignments also exist - Lawful Good people basically strive for the ideal of the benevolent philosopher-king, Lawful Evil people want to construct an oppressive and strictly regimented society, Chaotic Good people want to create an anarchist utopia, and Chaotic Evil people just want the world to descend into rampant murder and destruction.