I take an overly simplistic view of them, I know. And that's when I even use them.
Also, I've never written definitions down, or even summarised them in a like manner, so please forgive me if they suck.
1st hasty draft :
Lawful Good - Be either a hero or a saint (both if you prefer) and live up to that, but remember also to adhere to rules/laws/traditions/'normality' etc. unless circumstances dictate irrefutably otherwise, as these things are also absolutely vital for maintaining the greater good, which is Good manifested.
Neutral Good - Be either a hero or a saint (both if you prefer) and live up to that, and remember not to be overly swayed by such trivial issues as Law or Chaos. They can conceivably be 'throwaway tools' for a one-shot here and there, but in the end, all that is Good is all that matters
Chaotic Good - Be either a hero or a saint (both if you prefer) and live up to that, but remember also to defy rules/laws/traditions/'normality' etc. wherever possible, as these things cause stagnation, loss of liberty, stupor, ossification, and inevitably corruption (i.e. evil).
Lawful Neutral - Be ambivalent when it comes to good and evil; let those crazies fight it out...as long as they don't mess with the order of things, and as long as they adhere to the right way of doing so. Everything else should serve Law, too. If it doesn't, something is inherently wrong with it. Bring it into line or get it out of the way.
Neutral - Be ambivalent when it comes to good and evil. In fact, just be ambivalent. Stuff happens : it's the nature of things. Balance is essential, for without it life, the multiverse and all of us go insane. The multiverse is full of fanatics, sure, but frankly the fewer the better. Also, why waste time complicating life? Not complicated enough for ya? Feh, keep it simple stupid.
Chaotic Neutral - Be ambivalent when it comes to good and evil. Other than that, buck the system, ride roughshod over tedious, pointless restrictions. Be spontaneous!...if you like. Freedom is all that matters. Defend it at any cost, for without it why live? Unbind and unmake at whim. Liberate the minds of others from their needless shackles, when possible.
Lawful Evil - Be cruel and corrupting. Additionally, follow rules/laws/traditions strictly, as without adherence to Law, no dominion can succeed. Law is likewise upheld by the rule of evil, as that rule emphasises everything worthwhile - power, satiety, pain, slaughter - and those things enforce the rule of Law via ambition, greed, fear and bloodlust.
Neutral Evil - Be cruel and corrupting. Prepare to sacrifice anything and anyone else as needed to follow this simple maxim. Subjugate lesser beings to your superior will, and eradicate those who are more ah...stubborn, via whatever means. Have scorn for limiting Law and mindless Chaos. And hold an extra-special place in your heart for all that is Good.
Chaotic Evil - Be cruel and corrupting. Also, spit in the face of Law wherever it rears its predictable, sluggish head. Cut it off, perhaps. Power, satiety, pain and slaughter can all be achieved by the flexing of independent strength alone. So much more so, in fact. Never be tied down, never be predicted - Ha! Nothing will prevent your victory.