We just had a big alignment thread where I said my piece, but I can't help but post a little something on the subject.
I prefer to think of alignments beyond Neutral as being held by exceptional people and think that D&D in general has been too free about assigning them to NPCs. I also usually imagine that when a player says their character is Chaotic Good, for example, what they really mean is "Neutral, with Chaotic Good tendencies".
In the general sense, Chaos promotes possibility and Law restricts it, with the reasons for that varying based on whether they are pure or diluted by Good or Evil.
- A proponent of Chaotic Good believes that freedoms should only be restricted if it will almost certainly lead to evil. They believe that authority and hierarchy inevitably lead to oppression and advocate for a society where no one has authority over anyone else and all work for the collective good. They are optimistic about change and believe in the possibility of a society where selfishness does not exist.
- A proponent of Chaos values change for its own sake, no matter what the result may be. They oppose a stagnant status quo and seek to drive change for the emergency of novelty. They believe that Good and Evil ideologies are inherently limiting and impose restrictions on themselves and existence itself.
- A proponent of Chaotic Evil is concerned only with their own will and hates any limitations that would be placed upon it for the sake of others. True freedom for the individual comes from self-sufficiency and an ability to impose one's will on others not through outside authority, but one's own personal power. The ultimate goal of existence is to need nothing from anyone and to be able to do whatever you desire, even imposing upon others, without fear.
- A proponent of Lawful Evil believes that a strong, central authority and regimented hierarchy is needed to guarantee the greatest possible benefits for the individual through the influence and power that central authority invests them with. Personal advancement in the hierarchy is the driving purpose of life. Though one may currently dislike their place in the system and strive to achieve greater heights, authority must be protected at all costs if one wishes to protect their position and capability for promotion.
- A proponent of Law values certainty and conformity above all else. Everything must be standardized, regimented, categorized, and strictly controlled by a central authority. Good and Evil threaten Law by encouraging one care more about others or oneself more than the overarching authority. The system is more important than anything or anyone else; it must be dominant and eternal so that it can ensure continued stability and eliminate all uncertainties that could threaten order, and possibly existence itself.
- A proponent of Lawful Good believes that a strong central authority is vital to maintain order and create an environment where people need not struggle against one another. They believe that all people should conform to the same values lest a diversity of opinion lead to conflict. They value a time-tested status quo and distrust those who advocate for change without compelling evidence that said change will be for the better of society. Even when they themselves believe that change is needed to better realize the aims of Good, they are cautious and work through official channels to make that change happen, fearful that too quickly forcing change will result in unintended consequences that degrade society by bringing about conflict and uncertainty.
Lawful Good and Chaotic Good fundamentally oppose one another, with Lawful Good condemning the latter as dangerously reckless and foolishly optimistic, while Chaotic Good decries the former as valuing order over justice and being so pessimistic about change that they would rather existing evils continue than take a chance at improving things for the better.
Lawful Evil and Chaotic Evil fundamentally oppose one another, as well. Lawful Evil individuals value both the power and certainty that authority grants. For example, the devils of the Nine Hells believe in the rulership of Asmodeus and in the system he has created and invested with his own power. It is through the authority granted by Asmodeus that a devil can be certain they will be promoted if they perform well and that they need not fear being stripped of power or destroyed at a whim. In contrast to Asmodeus, demon lords cannot be trusted to honor their word or reward those they have cowed into servitude under threat of destruction.
Lawful Neutral and Chaotic Neutral oppose each other the most of all. At its furthest extreme, Law would eliminate all possibilities save for the one which is most ideal for its own certain, indefinite existence, eliminating free will in the process.