The Alignment System:
Now then, there are nine traditional alignments in D&D, though some people include a tenth. They are Lawful Good, Neutral Good, Chaotic Good, Lawful Neutral, True Neutral, Chaotic Neutral, Lawful Evil, Neutral Evil, and Chaotic Evil. The tenth is "Neutral Neutral", or as I will simply refer to it as: "Unaligned." Now, allow me to explain what the two axes of Law/Chaos and Good/Evil mean.
Good and Evil should be pretty straightforward. Being Good implies that your character is often altruistic, has a respect for life, will make sacrifices to help others. Evil characters are selfish, have little respect or value for life, and are oppressive. This has nothing to do with rules or regulations, however. The Good/Evil axis is simply a meterstick of how your character views the people around them. Neutrality on this axis implies you are neither compelled to help people in need, but you won't necessarily do them harm either. Generally speaking, Neutral characters on this axis prefer the company of those in the service of Good, because realistically, you'd rather have a Good neighbor than an Evil one.
Law and Chaos is a bit more murky. Being Lawful suggests your character follows authority, the law of the land, some personal codes of conduct (chivalry or bushido, for example). It also suggests that your character will keep their word, are trustworthy, and are probably honorable. Of course, Law has its downsides too. Law can lead to inflexibility, oppressiveness, and generally can keep an individual tied down into something that might otherwise harm them or people they know. Chaos celebrates personal freedom, flexibility, and choice. Chaotic characters are not necessarily bad, they just do not hold themselves to the law of the land and feel that the decisions they make for themselves are more important than what other people say. Chaos, of course, can lead to issues as well. Being Chaotic frequently makes you unpredictable, foolish, reckless, or disdainful to a legitimate authority. Neutrality on this axis means your character will generally do what they think is a good idea. They may obey authority sometimes, but they may not if they disagree with it at others.
Now that the axes are explained, the individual alignments are worth explaining.
Lawful Good: Lawful Good characters are frequently quoted as the "Knights In Shining Armor." This does not mean that all Lawful Good characters are this, however. Common traits of Lawful Good characters include compassion, loyalty, devotion to duty, honor, trustworthiness, and other such associated marks. They earn their lawful traits by adherence to a code, a cause, or the law of the land, and they earn their good traits by virtue of trying to do the morally right thing to do. They will usually refuse to break the law or their code in any given situation, instead intending to work with the system to fix things, preferring to let Justice be done. However, sometimes if a system or government is morally bankrupt or corrupt, Lawful Good characters can, and often do, rebel against it. Tyranny is the enemy of Lawful Good, after all.
Of course, Lawful Good does not necessarily mean nice. One might hold themselves to a morally strict code and always try to do the right thing, but they don't have to be kind, kitty, puppy and bunny-loving wimps either.
Neutral Good: Neutral Good characters are generally people who do Good for the sake of doing Good. They won't necessarily break laws or don't have moral codes, but they will not let them stop the character from doing what is morally right. Neutral Good characters act primarily out of compassion and concern for others or simply to move forth the cause of Good. Again, being Good does not necessarily make you nice, though Neutral Good characters often are. Even if a character is not fully sacrificing and altruistic, if they are interested in doing the morally right thing because that's what's right, they're probably Neutral Good.
Chaotic Good: Chaotic Good characters are individuals who frequently work outside the law entirely to further the cause of doing Good. They strongly value freedom for all individuals, not just their own personal freedom. They believe in changing the status quo for the overall greater good of all. Chaotic Good characters may break or ignore the law, but they're typically trying to do it for the right reasons rather than their own personal advancement. They're often likened to free spirits, but more interested in helping others or righting injustices, be they great or small.
Lawful Neutral: Lawful Neutral characters are typically concerned with the law, seeing it enforced, or holding it high. They are not necessarily interested in the moral quandaries associated with the Good/Evil axis. For the most part, Lawful Neutral individuals are about enforcing the law and seeing justice done regardless of the circumstances. Alternatively, they hold their own personal codes of conduct above that of what is traditionally seen as good or evil, though they are not especially malicious or oppressive about it. They might also be soldiers serving a nation and simply doing as they're ordered. It is important to note that their rigidity in following the Law does not make them bad or oppressive. They are simply the type to play by the rules and expect others to as well.
True Neutral: True Neutrality, for the sake of this alignment chart, is a bit of an odd bird. Those who are True Neutral actively seek to bring the forces of Law and Chaos and Good and Evil into equal balance. Such characters are rare and usually side with whomever is at a disadvantage in the endless conflict of these two axes.
Unaligned: Often referred to as simply "Neutral", Unaligned characters are... people who don't really feel strongly about the moral dilemmas posed by the Good/Evil axis, or the political and social ones caused by the Law/Chaos axis. For the most part, these are the people who simply live their everyday lives the best they can, doing what they feel is a good idea to get by. However, that is not to say that these people do not necessarily have any ideology at all, but rather do not feel the need to go out and prove it. Essentially, this is the "normal person" alignment. It is also the alignment given to many beasts and animals who lack the mental faculties to make decisions based on the two axes.
Chaotic Neutral: This alignment is for free spirits or people who are not beholden to the law and generally do as they wish. They may or may not care about the consequences. Chaotic Neutral individuals are most often concerned about their own gain, but they're usually not truly malicious in their doings. They may perform some underhanded actions now and then, such as stealing or whatnot if it suits them, but they're usually not the type to want to murder indiscriminately. As well, they tend to be concerned about freedom, but most of all they're concerned about their own. They also tend to be unpredictable, coming and going as their heart dictates.
Chaotic Neutral is perhaps the most frequently misplayed alignment in a D&D setting, with characters using it as a reason to get away with causing trouble, causing party infighting, and generally doing things that are frankly evil or more commonly, insane. I cannot stress enough that Chaotic Neutrality implies that a character does not enjoy undue suffering. And I cannot also stress enough that Chaotic Neutrality does not mean your character is insane and will do things completely at random. Though such things might fit the alignment, just because you are the alignment does not mean your character will do that.
Lawful Evil: Lawful Evil individuals are frequently likened to tyrants. Like the other Lawful alignments, they stick to a law of the land, a code of conduct, or something along those lines, but a key difference is that they are more likely to use their positions for their own personal gain. They are also very likely to bend the rules into their favor for the same reasons. They usually don't break these rules; they follow the letter of the law, but not the spirit of it. Often Lawful Evil characters aspire to power more than anything, and will seek to consolidate it in whatever means they can. It's worth noting that sometimes these characters display desirable traits. Lawful Evil characters will frequently obey orders and listen to the authority in charge, and because of this they may not go around committing atrocities if they can't get away with it or if it isn't smart. Also note that being Lawful Evil does not necessarily require you to murder, steal, or pillage. A Lawful Evil character can still be Evil because they don't care who they step on to get where they want to go and they're generally selfish.
Neutral Evil: Neutral Evil characters are potentially some of the most dangerous people in existence. They're not beholden to laws, but they're not as unpredictable or disorganized as Chaotic individuals tend to be. First and foremost, Neutral Evil individuals will stop at nothing to get what they want. They may be subtle or obvious in their dealings. Some Neutral Evil people actively seek to further the cause of Evil simply because they can. They are less concerned with ideology than they are with their own goals, wants, and desires. They have minimal compunction against inflicting suffering and injustices on others, though this is not a requirement. Some Neutral Evil people might not needlessly kill, in fact, many don't, because it might prove to be dangerous to them.
Chaotic Evil: Chaotic Evil characters are generally unpredictable, violent, selfish, and cruel. They value their own independence, but utterly are unconcerned with those of others. They will take what they want, do whatever it takes to get what they want. They will usually only fall into line out of fear of punishment. Generally speaking, Chaotic Evil characters are highly unapologetic about what they do, and the suffering of others is something they genuinely enjoy. Moreover, your character is not required to kill, but could be a sociopath who enjoys hurting people or performing cruel but random acts.
It is very important to note, however, that Chaotic Evil characters do not have to be berserkers or constantly performing evil acts. Many people misunderstand that about this alignment. It also does not necessarily mean your character is incompetent about performing evil acts. Arguably the best example of Chaotic Evil out there would be the Joker, from Batman, in his various incarnations. He's not always a murderer, but he's always Chaotic and he's always Evil.