I apologize for the wall of text.
All humans are born Neutral: there is a certain degree of selfishness in a Neutral alignment, there is a natural amount of cowardice and common sense, a willingness to be somehow a part of the current order of things. Neutrality allows also a fair degree of sentiments and love, of empathy and piety even. Loving a partner or a family member is Neutral, and so is hating their guts. Most humans are Neutral, and so is the human society at large.
Law is mostly a philosophical stance: when humans are concerned with more than themselves, they tend to a Lawful alignment. Someone in the hierarchy of the church is reasonably Lawful, as much as someone part of a cult, or of a secret society (assuming they believe in those organizations). Lawful is anyone who strives for some sort of order or purpose larger than themselves, either in a solitary endeavor or as part of a group with a common intent. Someone rebelling against nobility not for their own gain, but for their fellow commoners, is Lawful. And Lawful are probably the officers or nobles that fight to crush the rebellion, and strive to maintain the current order. Low-ranking rebels and common soldiers, on both sides, are probably Neutral, though. Lawful is an alignment for some, not for most humans.
It is clear, then, that Lawful has no Good or Evil connotation: it is the belief in something larger, in a purpose, in some sort of order, that makes the Lawful alignment be so.
Chaos, on the other hand, is the mark of everything non-human on a human soul. Using sorcery is chaotic and living according to the most simple rules of nature is chaotic (especially when the rest of the society does not). Chaotic is by no mean Evil. Chaos instead is in touch with the mystic energy of sorcery that subverts the rational laws, Chaos is a manifestation of a force (nature) larger than the single human life. But where Law brings order and has a purpose (regardless of what it is), Chaos simply is. It is capricious, it is fundamentally alien for humans, although some humans may indeed be touched and marked by it.
There is an order in Alignments; one must strive with consistency to bring themselves from Neutral to Lawful. Therefore, lack of such commitment resolves in a default declaration of Neutrality. The Referee may insist that a Lawful character change to Neutral, if they fail to prove their commitment with enough effort. Sacrifice (and often self-sacrifice) is required to maintain a Lawful alignment.
But as much as Neutral overrides Lawful for lack of commitment, Chaotic may override both Neutral and Lawful. Technically speaking, the usage of any form of sorcery is Chaotic. A lot of Lawful organizations maintain their lawful orientation and purpose, but any of their members that make use of sorcery are Chaotic (Magic-Users, yes, but I would say this includes Clerics too).
Someone might argue that magic, when blessed by the church for example, in the form of the clerical spells, is not Chaotic.
This is up to the Referee or the group to decide: if the Referee and the group can demonstrate that the higher source of power for clerical spells is god, then Clerics may be Lawful instead of Chaotic. This has a powerful implication: god itself must be Lawful if its magic does not bring the taint of Chaos with it. But if so, why is the world such a mess, why does not Law prevail on every aspect of our lives? What is the limit of the power of god, then? And is Chaos, and its sorcery, an enemy of a Lawful god, or are these simply two different entities, or even two different faces of the same divinity?
If the Referee and the group decide instead that clerical spells receive their power from the same source as sorcery, then Clerics must be Chaotic. Magic-users, of course, are always Chaotic.
(If you use demi-humans, stick to the default: Elves are Chaotic, Dwarfs and Halflings may also be Lawful, or more likely Neutral.)
Fighters and Specialist are usually Neutral and may strive to be Lawful, but how can they obtain the taint of Chaos if they wish to be Chaotic? May a simple curse do it? A mutation?
So, Neutral is the default state for humans; Lawful is an alignment which can be obtained through proper actions and a higher goal, while Chaos is a mark, an alignment brought by the power of sorcery.
Good and Evil perhaps exist, perhaps not. In gaming terms, Good should be treated as Lawful, and Evil as Chaotic. But this association is made by the human mind only. In truth, Good and Evil exist in these terms: they are strictly a consequence of the character’s actions.
Anything that causes harm that cannot be undone, is Evil. The scope of this definition is to be argued by the Referee and the group; is eating meat enough to be defined evil? Or is murder the line not to cross? What about murder with the intent of defending oneself, or even to defend one or more other persons? And where Evil is the harm that cannot be undone, it is therefore an Alignment that may very well be placed upon a character as a consequence of their action, even if the player is unwilling.
Again, there is room for argument here, and for house-rules: once the Evil Alignment is obtained, is there room for atonement and change? Or is this mark for life? Also, may the Evil alignment be associated with another, such as Lawful-Evil (not hard to imagine), Neutral-Evil and Chaotic-Evil? I do believe so. How to treat it in game terms? To me, Evil is predominant and therefore the character should be treated as Chaotic. But if the character for example is Lawful-Evil, they should be able to enjoy the benefit of the Lawful Alignment as well.
Good, on the other hand, has the simplest definition: it fights Evil.
It does not fight Chaotic or Neutral, and if it does, it will cause harm that cannot be undone and therefore become Evil. But as long as it fights Evil, Good may be treated as Lawful Alignment.
It is clear also that without Evil, Good cannot exist. It would simply revert to Lawful or Neutral or Chaotic.
Also, I stated previously that Good should be treated as Lawful, and Evil as Chaotic, and that this association is made by the human mind only. This is true: if the game inserts for example the idea of sin - in religious terms - as what defines Chaos and Evil, and religion as Lawful and Good, another character may argue that Good can be found only in the freedom of Chaos, and that any form of Lawful order is an imposition, and therefore Evil.
I do believe characters should be treated as they believe that they should be treated. You may have a character who claims to be Good fighting another who also claims to be Good; they both are, since they fight Evil in the form of the other. And both would reasonably end up being Evil, for the damage they'll cause.
Also, if a character claims they believe in the Good freedom of Chaos, so they want to be Chaotic-Good, the Referee may test them. If the Referee finds them hesitant, or catches a glimpse of guilt, perhaps the roots of the original sin have found their way into this character's heart, and they should instead just be Neutral, or simply Chaotic if they use sorcery.
The witch about to be burned at the stake is Chaotic because she uses sorcery, most of the crowd is Neutral, even those perverted enough to enjoy such a show. The witch-hunter is probably Lawful, but as soon as he burns the first witch, he becomes Evil, or better yet, Lawful-Evil. The one who betrayed the witch, and did it for a compensation in gold, not for a higher purpose, is simply Evil - this had nothing to do with Lawful.
The party of adventurers walking into the village right now are of different Alignments: there is a Magic-User who is Chaotic, there is a Specialist who is clearly Neutral, there is a Fighter who is Lawful and fights so that no one is burned at the stake, ever. This Fighter is not Good - too many deads behind - but is actually Lawful-Evil, same as the witch-hunter; just following a different path.
There is also a Cleric, level one, first time at the table, who believes to be Good (technically, Lawful-Good). At sunset, the Cleric will probably be Lawful-Evil or dead.
The Cleric might remain Lawful-Good, yes, as long as the corpses on the square, at the end of the day, are only the ones of those who are Evil.
So, this was a personal take on the LotFP alignements system, but if you want to read some more about alignments, you can check this post of mine: Alignments.