Lawful Good
Often disdained as the paladin's goody-two-shoe alignment, Lawful Good is a far more subtle alignment than that, for very few characters who happen to be Lawful Good actually stick to anything approaching the Paladin Code in stringency.
A character who is lawful good holds strongly to the principle that the good of the society can only be maintained via order. The exact specifications of what is "good" may vary wildly for this purpose, but on the whole "good" implies concern for the safety and comforts of others, and that all persons have a right to safety and comfort. The character may build up other definitions in his own moral code, but this is essentially the heart of the matter. Lawful Good, unlike the other Good alignments, also holds that it is sometimes necessary to sacrifice what is "good" for a few persons in order to preserve order, which is "good" for society in general.
On a personal level Lawful Good characters have a tendency to at least strive for an ordered lifestyle. Many, if not most of them, have an "everything has a time and place" attitude. Most tend to organize their behaviors to suit the group, although they are reluctant to do so at a personal loss unless something worthwhile can be obtained with the sacrifice.
A Lawful Good alignment does not rule out chaotic behaviors, particularly impulsiveness. Impatience is another fault that some Lawful Good characters are guilty of. On the other side of the coin, hedonistic behavior is not often observed in Lawful Good characters, unless in one or two activities.
Lawful Good characters believe that order is the best way, they don't necessarily maintain it. A character with a particularly poor willpower may allow his life to get quite out of shape. And also remember that the character will maintain order from his point of view. A Lawful Good merchant's shop may look like an explosion in a mattress factory, with nobody other than the character able to find anything. But the apparent chaos is deceiving most of the time, ask him where something is and he'll usually be able to find it much faster than appearances would seem to dictate.
Lawful Good societies tend to avoid overdoing laws. The laws of the society are usually well coded, but one individual, or sometimes a council, has the right to overrule the law to achieve something "good" (or at least perceived as such). Lawful Good justices are more concerned with the spirit of the law and the intent of its author in achieving a worthy purpose. If the law does not achieve this most Lawful Good characters will dispose of it, although some more reluctantly than others.
As a DM running a Lawful Good PC, be careful not to be paladin strict on them. While it is true that paladins must be Lawful Good, the reverse is not true. Some Lawful Good character may take active disagreement with the Paladin's Code, let alone try to live with it. Most Lawful Good characters respect the Paladin's Code, and respect members of the class for trying to live up to it, but they hardly make such an attempt themselves in their day-to-day lives. For the vast majority of Lawful Good characters the occasional charitable act is enough, unlike paladins they usually won't go out of their way to help someone. But they will never go out of their way to harm someone either - indeed most Lawful Good characters that do this accidentally will try to rectify the situation as soon as possible.
Neutral Good
It is the belief of most Neutral Good aligned characters that morality must be maintained beyond all other virtues. As long as someone is not harming others then he is doing nothing wrong. Neutral Good characters also perceive both too many laws and too few of them to be a threat to the common good. Too many laws and freedom will be unjustly restricted, too few and it will not be protected. Hence, a balance must be struck.
Neutral Good characters strive to act to the benefit of as many persons as they can, and if that includes themselves that is an added bonus. If they cannot personally benefit from a good act they will still probably perform the act. They will not intentionally harm anyone who is not a known enemy, and such dealings are usually short and to the point. Neutral Good characters are perhaps the least tolerant of whatever they perceive as being "wrong" of the three major good alignments.
Individuals of this alignment can be quite interesting. Not overly concerned with establishing order in their lives, they may have the appearance of being chaotic, yet unlike chaotic characters they rarely rebel against a just authority simply because it is overbearing. Although it varies greatly from individual to individual, Neutral Good characters do not usually rock the boat. A Neutral Good leader is rare, for the individuals of this alignment often lack the sense of duty or the sense of individualism it requires to take on such a task.
As is mentioned previously, there is a balance between the lawful and chaotic sides of a Neutral Good character, although it should be noted that the vast majority of those characters do not consciously maintain this balance in their lives. It is this lack of a conscious awareness of the balance that may explain why there are so few Neutral Good characters when compared to Lawful or Chaotic Good ones: they simply fall to one of the two waysides.
An entire society made up of Neutral Good characters is rare, although it should be noted that the gnomic society bears this trait. Such societies are usually quite small. Government is achieved through a few leaders and a few, broad edicts that everyone is happy with. The happiness of the greater part of society is the only real concern, any laws that would affect something else are usually not to be found.
Dungeon masters that run a Neutral Good character should pay attention to the balance between law and chaos in the character's general behaviors, but don't be to quick to penance aberrations with the members of this alignment. A Neutral Good character may follow every order given to him faithfully until he doesn't agree with one, at which point he simply leaves. When gauging alignment shifts, look for chaotic and lawful actions whose natures precede over the greater good. For instance, a paladin may follow (or at least capriciously twist) a questionable order, a Neutral Good character would ignore it altogether. A Chaotic Good character may disobey a just order because he doesn't agree with it, even if he's been proven wrong. Neutral Good characters will concede (usually) in such cases.
Chaotic Good
The Chaotic Good mindset is a paradox of the desire for individuality and a benevolent concern for others. Chaotic Good characters are almost always strong minded and each tends to possess a unique moral compass about what exactly is right and what is wrong. These characters have a tendency to pass moral judgments based on their beliefs, and to hell with what anyone else thinks. Despite this individualism, the definition of good, for most Chaotic Good characters, is don't hurt anyone else.
Chaotic Good characters tend to be popular among players since it is mistakenly viewed as the only good alignment with room to slip up once in a while. That isn't true, for all good alignments can waver quite a bit. Chaotic Good characters, consciously or not, resist authority because they have a tendency to equate law with evil. They have the universal precept that every man should be left to his own to make up his own mind. In this aspect Chaotic Good characters can be very anarchistic, since they have a trust that people are capable of governing their own behavior.
There is a lot of variety in Chaotic Good behavior. Some are hedonistic, others have a paladin-like piety, but lack the follow through of the members of that class. Most are restless however, some are meticulous. Just as it is a mistake to believe Lawful Good characters to be devious of chaotic streaks, so too is it a mistake to believe Chaotic Good characters to be incapable of order. Indeed, often the characters of this alignment have to follow orders, even when they don't want to, but they can be quite capricious in their dealings with authorities they do not like. On the other hand, they can be quite methodical in order to get what they want. Still, the chaotic mindset is there. Chaotic Good characters who fall into a routine will break it occasionally for no apparent reason to anyone but themselves.
Chaotic Good societies tend to be wide spread and lacking of any real central authority. These societies rarely have a set of laws set in stone, just a series of precedents and understandings. "Rule" per se, is often deferred to one person at a time. Large Chaotic Good societies will enact more laws, but they will still be very broad based. Often these societies are republics with a democratic (or at least semi-democratic) leadership elected from time to time. But no matter what the intentions of any proposed law in such a society, it will always have opposition from those who feel that too much government is wrong, or even evil in nature. Enacting any new law in such a society is often a long and laborious process (If all this sounds familiar, the United States is an example of a Chaotic Good society and government).
DM's running Chaotic Good characters should take care to watch the player's lawful leanings. If he is too cooperative too much of the time, he's probably playing Chaotic Good because he things that Lawful (or Neutral) Good is too restrictive. On the contrary, Chaotic Good can be just as restrictive in the sense that it details a certain type of mindset and society. Still, don't be too harsh on this or any alignment. Even a Chaotic Good character will not rebel without any reason, although said reason may be whimsical at best.