Lawful Neutral
In a sense, Lawful Neutral characters can be the most dangerous of all the character types, especially in large numbers simply because, for the most part, they are easily controlled. Lawful Neutral characters place order above all other concerns. To them the presence of a natural order is the way things must be, and any moral issue stemming from the use of order is completely irrelevant to them.
Lawful Neutral characters look for patterns in everything. They cannot accept the world as being chaotic, and they therefore devise systems to explain away anything chaotic in their lives. It is chaos that they fear more than anything else, a Lawful Neutral would endure a thousand tyrannies before living in an anarchistic situation.
The characters of this alignment tend to always follow rather than lead. Leaders in Lawful Neutral societies are almost always Lawful Good or Lawful Evil, for rarely does a character gain a position of preeminence without gaining some moral baggage along the way. Even when Lawful Neutrals do lead, they follow whatever laws and precedents that preceded them. Lawful Neutrals are notoriously lacking in initiative, but to say they must have no willpower or individuality would be a mistake. Although they will fit in quietly and nicely when the powers that be follow the letter of the law, they can be a very painful hindrance to those who would try to follow the spirit of the law (Lawful Good) or twist it to their own ends (Lawful Evil). Order for its own sake - Lawful Neutral characters will stubbornly defend this principle to the bitter end. Just as there are few Neutral Good characters, so too are there few Lawful Neutrals. Most of these characters fall to the waysides of Good and Evil before holding their alignment long.
Lawful Neutral societies are almost always large, and always come with large cumbersome (and despite all efforts, usually comically ineffective) bureaucracies. Everyone in such a society is assigned a place in life. The goal of a Lawful Neutral society is to achieve perfect harmony, where everybody has a job, a place to live, and through the benevolence of a well run government everyone is happy (communism is an example of a Lawful Neutral governing system). Unfortunately, the willingness of these societies to follow orders and laws proves to be their undoing, as inevitably a few evil persons become entrenched at the upper echelons of government and are virtually invulnerable from removal or attack.
DM's should be wary of Lawful Neutrals. As with Chaotic Neutral it is an alignment that rarely lasts over a character's career. Lawful Neutrals are only occasionally interested in self-advancement unless they feel that their advancement (via their talents) will benefit society as a whole. "The needs of the many outweigh the needs of the few" is the Lawful Neutral creed, no character of this alignment will go long without sacrificing material possessions, themselves, and even their friends for what they perceive to be for the benefit of their society.
True Neutral
True Neutral is at once the rarest and the most common alignment in the multiverse. Neutral characters are either incapable of making moral and ethical decisions or refrain from them actively. "Actively" is the key phrase here. A character who is incapable of moral and ethical decisions would have an intelligence and wisdom scores no higher than 6 each. Very few player characters qualify on that account, therefore the vast majority of PC Neutrals actively refrain from moral and ethical judgments.
Within this context there are two reasons a character may not actively pursue ethical and moral thinking. The first, and most storied in the AD&D tradition, is the druidical "preservation of the balance" mentality. However, this view, for obvious reasons, is uncannily rare.
The second and most common example is the character who simply doesn't care. Neutrals of this type aren't concerned for the welfare of anybody (except themselves), but they aren't willing to advance themselves at any cost like Neutral Evil characters. They are comparatively lazy and lacking in drive in this respect, still there are certainly other reasons. The character may see advancement beyond reasonable comfort through reasonable risk as pointless. Not surprisingly, these Neutrals have a bleak view on life. As for the ethical (law vs. chaos) leanings of these characters, they may follow orders if it serves them, disobey if it doesn't, act on the occasional whim but otherwise keep to a reasonably predictable life.
True Neutrals are all about reason, and perhaps that's why they are so few. Everybody seems to get behind some banner or cause, Neutrals couldn't care less about such trivial matters. Often they have one fascination in their lives which they are preoccupied with, and everything else doesn't matter to them. In the case of druids, this is the "preservation of the balance."
True Neutral societies are very rare, but when they do occur they tend to be primitive. The concerns of simple survival dominate the daily lives of such people, and for them further complications aren't worth the effort to create. In a way, Neutrals are the most peaceful of societies, but they are also the most infuriating because of the nearly universal lack of a drive the members of this alignment possess.
DM's running a True Neutral should watch for any consistent behaviors that lean towards another alignment, but often he will find that there is a balance between the various aspects of the character's life, whether he consciously maintains them or not.
Chaotic Neutral
Perhaps the most misunderstood of the nine alignments, many people believe that all chaotic neutral characters are insane. While it is true that none of them are very predictable, insanity is not the province of the members of this alignment alone. Indeed all characters of all alignments are capable of insane and compulsive behaviors, but Chaotic Neutrals are not only the most inclined to this type of behavior, they are also the most famous for it.
Chaotic Neutral characters have little to no discipline. That is perhaps the only statement that can be made about this alignment that applies to all the personalities contained within its context. As a rule, the only concern of Chaotic Neutral characters is themselves. They seek to do what pleases themselves and don't really care if that pleasure includes anyone else. Chaotic Neutrals can also be surprisingly snug and unambitious. This lack of personal greed separates this group from Chaotic Evil characters, who are often very concerned with getting ahead. Chaotic Neutrals are also unconcerned with philanthropy in the least. They will help someone that it pleases them to, or someone who will pay them.
Chaotic Neutrals do not consider morality in their judgments, and about the only thing they stop to consider is whether or not they have a remote chance of success. The odds aren't usually considered, but even Chaotic Neutrals will not attempt the obviously impossible unless desperate (then watch out, anything could happen). Chaotic Neutrals aren't suicidal, neither do they shy away from trusting sheer luck, and hence the alignment has reputation for brash behavior.
Chaotic Neutral societies are anarchies. Everyone does pretty much what he wants, and therefore Chaotic Neutral societies are either very small, or do not last long at all. The only real example of such a situation are the Mad Max movies and others in the post-apocalypse genre. The very lack of authority invites its institution, usually by force of arms.
Dungeon Master's need to watch this alignment carefully. Unscrupulous players sometimes play this alignment instead of Chaotic Evil if the DM bans that alignment from play. Outside of that, there is an oddball balance between good and evil in the personalities of Chaotic Neutrals. They may assassinate a good and benevolent ruler for cash, then spend the money on the poor. Although the contrast is rarely that sharp, it will be there when the alignment is played right.
It is also necessary to remind all that Chaotic Neutrals aren't insane, or crazy by definition of alignment. Chaotic Neutral characters as individuals may not take high risks, especially intelligent ones. However, they have a tendency to take odd approaches to objectives. Chaotic Neutral characters also thread intricate plots, but unlike lawful characters these plans are flexible and can change as whim and situation demand.