My son Billy is a keen Dungeons and Dragons player and game-master; and when he was trying to explain to me how the Alignment system worked, he devised an example using Tolkien's orcs - which ended-up by throwing light on the different nature of the orcs themselves.
Lawful Evil - Saruman's Uruk-Hai
Neutral Evil - The orcs (or goblins) of the Misty Mountains, including Moria. (These are depicted in most detail in The Hobbit.)
Chaotic Evil - The ordinary rank-and-file Mordor orcs
The Alignments of evil can be differentiated by how selfish the orcs are, how short their time horizon; and by how much good is mixed with their evil qualities.
The three Alignments can be seen played-off against each other in the Uruk-Hai chapter of The Two Towers, when representatives of all three alignments can be seen contrasted, as a mixed companyof orcs carry Merry and Pippin across the fields of Rohan.
The Uruk-Hai are Saruman's elite warriors, cross-bred by the wizard from Men and Orcs. They are Lawful Evil because they have Good qualities of loyalty, obedience and courage - which are directed-towards evil goals.
The Uruk Hai are personally loyal to Saruman, whom they worship; seem to lack selfishness and personal ambition; and they follow Saruman's orders as an abstract Good - which gives them satisfaction (and not just because they fear being punished if they were to disobey).
Thus Ugluk ensures that Merry and Pippin are healed of wounds, fed, and given an energizing draught; so that they have the strength to run long distances and can be delived to Saruman quickly and in good condition; according to his orders. He also, for the same reason, protects his prisoners against the Chaotic Evil Mordor orcs - who merely want to eat, despoil and torture the prisoners.
In a sense, Lawful Evil is evil 'only' because it is harnessed to evil purposes; and because it denies any personal responsibility - being ruthless in pursuit of obedience and loyalty.
The paradox is that lawful evil is by far the most effective type of evil (just as the Uruk-Hai are by far the most effective orc troops); but that its effectiveness is a consequence of the evil being mixed-with good.
At the opposite extreme from the long-termist, self-effacing obedience and loyalty of the Uruk-Hai, are the rabble of Mordor Orcs. These are a much 'purer', unmixed form of evil.
They delight in inflicting pain, laughing at misery, and in killing - and are just as happy to torture and cannibalise orcs as anyone else.
In a sense, Chaotic Evil orcs are impulsive psychopaths; liable at any moment to fight each other to death, and abandon the mission and ignore orders to indulge in a spree of looting, arson, murder. They will go to considerable efforts to destroy; simply because they enjoy destruction. Any mission can be sabotaged (hostages beaten, tortured, killed, dissected, eaten) simply because they become overwhelmed by the immediate desire to do harm.
It is presumably these orcs who chop down trees and leave them to rot, pollute their own drinking water, and burn crops. They are like virulent parasites who kill their own hosts, then each other - and then die themselves.
Chaotic Evil orcs are only ruled by the immediate threat of punishment for disobedience; or by the kind of coercive mind-control that Sauron or the Nazgul Witch-King can impose upon them. Left to themselves, they are useless - and self-destroying.
The orcs of the Misty Mountains seem to be exmaples of Neutral Evil, as is Shagrat and, to a lesser extent, Gorbag who are captains in Cirith Ungol. Their moral nature lies between Lawful and Chaotic evil - with aspects of both; but Neutral Alignment has a self-sufficiency which is its own.
The Hobbit depicts a goblin (orc) civilization in the Misty Mountains - in which individuals are loyal to their orc king (Bolg); and cooperative to the point of risking their lives in pursuit of his killers (i.e. Bilbo, Gandalf and the dwarves); eventually waging war upon them (in the Battle of the Five Armies).
Neutral Evil Orcs seem to have a clannish, gangster-like society with a medium-term time horizon; able to cooperate in construction of the underground chambers and tunnels and invention of weapons and machines; wanting mainly to be left alone to live by relatively low-key theft, piracy, kidnap and the like. One can imagine them running protection rackets with the surrounding Men.
Their evil is in their means employed, more than their ultimate ends. In terms of their virtues - they are loyal, but not lawful; they are reasonably obedient - but it is easily overwhelmed by temptation. They lie without conscience, even to their leaders; when they think they can get away with it - and pretend to the same kind of virtues as Men; but purely expediently.
Neutral Evil orcs do not seem to be motivated by conquest - except when their numbers increase and they need more resources.
This aspiration of Neutral Evil orcs is spoken in the conversation Sam overhears between the Mordor orc-leaders Shagrat and Gorbag:
[Gorbag]: 'the war's on now, and when that's over things may be easier.' `It's going well, they say.' 'They would.' grunted Gorbag. `We'll see. But anyway, if it does go well, there should be a lot more room. What d'you say? - if we get a chance, you and me'll slip off and set up somewhere on our own with a few trusty lads, somewhere where there's good loot nice and handy, and no big bosses.' 'Ah! ' said Shagrat. `Like old times.'
The nostaligic tone for the good-old-days when there were 'no big bosses' - when orcs ran their own affairs - is unmistakeable!
While Lawful Evil orcs live to serve, and ask for nothing more; and Chaotic Evil orcs serve to live (or else they will soon kill each other); Neutral Evil orcs crave their own version of an autonomous Good Life - The Orcian Dream...