Caveat: This is a topic that has been, is being, and will be argued for time immemorial. So YMMV, of course.
I think of “Lawful” as a pattern of behavior that is organized and relies upon rules and systems to make things work. Discipline and “the rules” are how lawful people live their lives. Lawful people tend to be methodical, rigorous in sticking to a routine, and follow a very specific pattern of how they do what they do. Sometimes this makes them predictable, though sometimes they are very adept at outside-the-box thinking that can surprise opponents. While that may seem like a fundamentally chaotic thing, it only appears that way to an outside observer: a Lawful Neutral bounty hunter’s MO, for instance, might always include finding new and innovative ways to surprise a mark.
I’m going to give examples from comics, video games, and my own books to exemplify these alignments–note that these aren’t necessarily perfect examples, as many of these characters have had countless iterations and visualizations and you can argue lots of exceptions. A lot of these characters (particularly the LN ones) have good or evil tendencies, and that’s fine. In the case of Geralt, for instance, choices you make while playing the games he’s in can push him in a good or evil direction–he isn’t strictly neutral. These characters are sentient creatures who aren’t uniform in their behavior. Alignment isn’t a straight-jacket–it’s a general tool for describing behavior and outlook.
A Lawful Good person believes in law and order being tools for the benefit of all, and will follow the laws of the land so long as the higher ideal of justice is served. A Lawful Good person has a strong sense of compassion and prioritizes helping those in need, even if it’s dangerous to do so. They are often extremely driven people, unable to tolerate injustice or stand by and do nothing.
Lawful Good types will be extremely uncomfortable with the very concept of bending the rules, much less breaking them, even if it’s for the greater good, in a way that a Neutral Good person would not mind as much, while a Chaotic Good person would advocate for breaking oppressive rules as the best course.
Examples: Superman is Lawful Good. Obi-Wan Kenobi is Lawful Good. Daredevil is Lawful Good. Triss Merrigold is Lawful Good. Kalen “Shadowbane” Dren is Lawful Good.
A Lawful Neutral person believes in law and order for their own sake, basically “those are the rules and we should obey them because they’re the rules.” This doesn’t necessarily mean that a LN person always obeys the laws of the land they’re in, particularly as a traveler, but they always have a set of strictures or a code that they follow to the letter, and generally they default to a basic respect for the laws of the land, as in “those are the rules that others follow, and they follow them for a reason.” They are often thought of as mercenary or “just following orders” types.
There is a hierarchy of rules: a Lawful Neutral person will only violate an existing law if it comes in conflict with a more important law, and then usually with great discomfort. They may or may not put on the appearance of being good and are sometimes described as being amoral or unfeeling (which is sometimes accurate).
A True Neutral person doesn’t cleave to the law in this way, while a Chaotic Neutral person may have similar priorities to a Lawful Neutral person (getting paid to do a job, for instance) but goes about it totally differently, ignoring or violating rules and expectations as a matter of course.
Examples: The Punisher is Lawful Neutral. Mace Windu is Lawful Neutral. Dexter Morgan is Lawful Neutral. Gerald of Rivia is Lawful Neutral. Levia Shadewalker (Shadowbane 3) is Lawful Neutral.
A Lawful Evil person believes in law and order as a means for securing their own power and dominance. The rules are important, primarily because they can be exploited to disadvantage others. A Lawful Evil traveler pays only lip service to the laws of the land that conflict with their own personal code and set of strictures, and will ignore those laws they consider to be weaker than their own or worthless. A Lawful Evil person seeks power through organization and alliance, relying upon others to provide them the support they need to achieve their goals, which involve crushing their rivals.
Neutral Evil people may take advantage of laws but don’t feel much compunction about violating them or working outside them at the drop of a hat, while Chaotic Evil people usually revel in defying laws and rules and will gleefully shirk them whenever possible.
Examples: Lex Luthor is Lawful Evil. Two-Face is Lawful Evil. Ra’s Al Ghul and his daughter Talia are Lawful Evil. The Red Skull is Lawful Evil. Doctor Doom is Lawful Evil. Darth Vader is Lawful Evil. Vengeance (from Shadowbane 3) is Lawful Evil.