I think the whole alignment system is the worst aspect of D&D, being retardedly simplified and restrictive (especially if you've got a DM who's strict with the rules), but as a role player I do of course have my own view of the alignments:
Lawful Good: Law and order for the good of all.
Lawful Good characters value law and order over everything else. Like all Good characters they honestly want what's best for everyone. They just strongly believe law and order are necessary to make a world where everyone can be happy, and that there must be a strong force to preserve that order, with extreme prejudice if necessary. After all, people cannot live in peace if their safety cannot be guaranteed, and as long as there are people willing to hurt others for fun and profit, there must be a powerful presence of order to protect the innocent. He values order over almost everything else, but mostly out of a strong belief that in the end it's the best thing for everyone. While the nature of such order can be almost anything, I do believe that it has to be something external, be it a law or a deity's commands or a knightly code. Just a personal code of conduct isn't enough to qualify as Lawful Good, otherwise Robin "steal from the rich, give to the poor" Hood wouldn't be the iconic Chaotic Good character.
Neutral Good: Anything for the greater good.
A Neutral Good character just wants to help. Honestly. They want everyone to be happy, and they don't care how this is achieved. One day they'll uphold the law to protect the innocent, and the next they'll break the law to feed the poor. This isn't out of randomness, but simply because to them everything is secondary to doing good. They'll do anything to minimize harm and maximize the amount of people they can help. They don't necessarily mind breaking some skulls if there's no other way, but they prefer a solution where everyone walks away happy. Of course, such solutions are in short supply in most role playing games.
Chaotic Good: Freedom is the greatest good.
Like all Chaotics, the Chaotic Good character has a deep aversion for laws and regulations. They believe such rules only serve to oppress people and limit the freedom which is every sentient being's right. He believes the world would be a much better place if people would just stop bothering each other and live their lives the way they want to. Of course, he's usually not naive and realizes that there are people who would abuse such liberties, but he prefers to deal with them on a case-by-case basis and on his own terms, rather than relying on laws to prevent such behavior.
Lawful Neutral: There must be order.
Like the LG archetype, the Lawful Neutral character believes strongly in law and order. However, unlike the other two Lawful alignments, this order is not a means to an end for him. To him, order is a goal of its own, perhaps a more important goal than anything else. Be it out of a sense of honor, duty, obligation, religion, or anything else, the Lawful Neutral character will follow the law to the letter, and expects no less from others. As always, the source of this law might be different from character to character, but his rigid and uncompromising attitude towards it does not.
True Neutral: Balance in all things, or I just don't give a shit.
I feel there are two types of Lawful Neutral. The most iconic is the version most commonly ascribed to D&D Druids: There is a reason for everything, and the world must be in balance in order to function. They believe that Law, Chaos, Good and Evil are all necessary parts of the natural world, and that none is inherently better or more desirable than the rest. They usually prefer a state of peace and inactivity, but they will rise up and take action when they feel the balance is threatened or disturbed.
The other kind of True Neutral is a more passive approach to alignment. They are the people who really don't care about the different alignments. They do what seems like a good idea at the time, but have no real preference for malice or compassion, order or freedom. They act mostly in their own interest, but without getting overly selfish or greedy. They can be mean or kind at times, and they can be obedient or anarchistic, not out of any kind of personal conviction but because it happens to suit them at the time. Non-sapient creatures have this alignment by default.
Chaotic Neutral: My way is the best way, for me.
Allow me to start by saying that I really, really dislike the "lol I'm random and unpredictable and kind of crazy~!" kind of Chaotic Neutral. To me, Chaotic Neutral characters are all about freedom. A Chaotic Neutral character is (not surprisingly) a mix between the TN and CG alignments, in that he shares the CG's love for freedom and dislike for laws that limit such freedom in any way, but also has the TN's disregard for anything but himself. He won't help you get your kitten out of a tree, but he won't beat you up for your lunch money either. He's got a very strong individualist streak, but that doesn't necessarily make him a loner. He lives for his own enjoyment, and it's not surprising if that enjoyment means being around other people. It's just that he won't inconvenience himself to help others, nor will he try to harm them in any way if he can avoid it. That doesn't mean he can't do good or evil if the fancy strikes him, but it's not in his nature to go out of his way for either one, and he'd rather just let people do their thing as long as they let him do his.
Lawful Evil: Law and order for the good of... me.
The Lawful Evil character believes in order as a tool to be used not for the good of all, but for his own benefit. He wields laws as both his weapon and his shield, and manipulates them in such a way that he always comes out on top. He uses the law to get what he wants, and when in trouble will hide behind the law in order to protect himself. After all, others might suspect or even know he's up to no good, but good luck trying to prove it if he never actually broke the law. Lawful Evil characters are often patient and skilled deceivers, manipulating others to do their dirty work for them. That doesn't mean they won't use violence themselves if necessary, but they prefer to let the law do their work for them, with a little help here and there to nudge the proceedings in the right direction. and by "the right direction" I mean "whichever direction is most convenient for them".
Neutral Evil: The more I let you have the less that I'll be keeping for me.
Neutral Evil characters are selfish to the bone. There's nothing they wouldn't do to benefit themselves, and other people are nothing more than potential ways to improve their own situation. They might not usually go out of their way to harm people all the time, but they won't hesitate to kill a person for a bit of money, or even just for fun. They don't see the use in helping other people unless there's something in it for themselves, and even then they are more likely to kill the farmer and take his gold rather than go kill the rats in the cellar for a reward, if they thought it'd be quicker and/or easier. They might not care much for laws and regulations, but they don't have any real trouble with them either as long as they don't conflict with their own interests.
Chaotic Evil: Fuck it all, fuck this world, fuck everything that you stand for.
Chaotic Evil characters are the psychopaths. Like all Evil characters they are utterly selfish, but unlike the others they lack the ability to function in an orderly society. They are wild and unpredictable, able to kill a man just because he looked at him funny, and without any regard for the consequences. The only 'society' they are able to function (and perhaps even thrive) in is an animalistic place where they can exercise the right of the strongest. Other than that they are loners, shunning others as they are shunned by them. Other people are nuisances at best, and squishy things that make funny noises when you gut them at worst. They are completely unable to abide by any rules other than their own violent urges.