Two things about alignment are important for the DM to bear in mind are that it is an absolute system invented for the purposes of the game, and it is not voluntary.
The implication of the first point is that the characters' opinions of what is good and evil are not important. No evil person would declare themselves to be "evil". They may rather scoff at the idea of being evil or good and simply say that they are strong while those that call them evil are weak sheep bleating at the wolf who simply has the strength to take what he wants.
The implications of the second point - that alignment is not voluntary - are rather cosmic and we'll come back to them later in the post.
First, the definitions of what the alignments mean in this absolute sense. There are two axes, with two poles on each. The Ethical axis has the poles "Lawful" and "Chaotic", and the Moral axis has the poles "Good" and "Evil". There are many such poles and axes in AD&D, reflecting a late mediaeval influence in the cosmology, but none are so important.
Good is primarily concerned with life and the quality of life for all, where possible. Things which create and grow healthy living things and allow them to be free from fear, disease, injustice, oppression and so on are "Good".
Evil is primarily concerned with dominance and strength. Those things which take from the many and give only to the few and which cause needless suffering and malicious, non-accidental, pain and loss are "Evil".
To act in concert, to bind together in order that the sum of the result is more than the parts. To subsume the individual's desires into the desires of a group so that the group can attain more than the individual could ever hope to, is "Lawful".
To act as one's own conscience says, to ignore the expectations or demands of others to conform. To put the purity of attempting to attain ones own goals above the value of compromised success within a group or society, is "Chaotic".
Combining these traits together leads to the 8 major alignments, and the neutral spot in the middle makes the classic 9. Further shading of the 8 major alignments into each other gives the standard view of the Outer Planes with their 16 uppermost planes (including Concordant Opposition, introduced in Deities and Demigods and not actually part of the core cosmology in PHB).
Neutrality comes in three forms: the character who purposely equally engages in actions which fit both ends of an axis; the character who never does anything especially biased towards any extreme, and the character who "does whatever works" in any particular situation without regard to dogma. The latter is the least stable position to be in if the character is of any note (ie, high level).
The first form of neutrality is called "true neutrality", although the term is normally reserved for a character who takes this stance in relation to both axis, not simply a lawful-neutral or neutral-evil character.
Notice that there is nothing explicit in "Law" about obeying laws. "The Law" is, as they say, a human institution, whereas "Law" is a more abstract thing. A Lawful Neutral knight can safely ignore any law or edict passed by, for example, a usurping king or evil high priest. If s/he has made an oath, however, then breaking that might be a lot trickier.
Remember when I said that the alignment system is involuntary? Let's look at that a bit more. Key pages from the rules here are DMG p80 (Saving Throws), and 81 (Hit Points) and PHB p34. Notice the references to "supernatural powers" in those sections? This is where alignment comes in.
In essence, AD&D's "big picture" is of a chess game played out chiefly in the worlds of the Prime Material Plane. Somewhere out there, beyond the gods, are supernatural powers, perhaps unthinking laws of the multiverse, perhaps something else. Whatever they are, they vie with each other presumably in an effort to make their pole of their axis the ruling one, extinguishing the other.
If the character does things which thwart the goals of the alignment, then eventually they will be "dropped from the team" and lose those protections. Fortunately, leaving one team means joining another, since they are opposed. Even going from LN to N means that the character is becoming more useful to Chaos. As a result, changing alignment only results in the loss of one level, and that only if the character is 3rd level or higher at the time. See DMG p25.
A character who is in danger of changing alignment involuntarily, can atone (PHB p49) but we'll leave that for a separate post; it's not important at the start of play.
If you, as DM, look at the actions of the characters and simply ask yourself which of the four poles would be "pleased" by their actions, then alignment questions largely become trivial. Some will balance out, some will move the character towards one or more poles. How far any particular action moves them, and how much movement is needed is up to you and it's probably not worth reducing it to a hard and fast numerical rule which becomes another rule to remember.
Which is handy, because there is one other aspect of alignment which the starting DM does need to be aware of - performance ratings.