The most common NPC interaction is with the player characters, usually requiring a response from the PCs. A villain wants to belittle or harm the characters. Some NPCs provide information. Some are catalysts for starting an adventure and point the characters in the right direction. This is all about how the NPC directly interacts with the PCs.
The second most common NPC interaction is with themselves. These are self actions (or quirks) an NPC will take that do not directly require the PCs (or anyone) to respond to the action. An NPC might hum when he gets nervous. He might have yellow hair. He might have a lisp. She might be a dwarf who walks around on stilts.
Having an NPC interact with another NPC is an excellent way to show the personality of an NPC. However, as a quick note, I am not advocating having NPCs engage in long discussions with other NPCs; that is counter to putting the players forward in a game and should be avoided at all costs. Instead, there are ways to quickly show how an NPC interacts with other NPCs without taking the spotlight away from the player characters, sometimes without even involving the other NPCs.
I have a few examples…
One obvious example is the villain. He can be seen to do things to other NPCs, both as a way to show how powerful he is and also to show how evil he is. A villain fireballing an NPC’s home is one form of this type of NPC to NPC interaction. Cutting down an NPC minion who has failed him is an easy way to show how callous he is. Monsters who have defaced a holy temple show their disdain for that deity and the things that deity represents.
Hirelings and NPC companions are often given a variety of quirks. Most are just to add color, but by having their quirks include other NPCs it makes them more “real”. Having a torchbearer constantly talking about his wife and 14 children (“that gelatinous cube reminds me of when Callie, that would be my third daughter, had a cold and coughed up this huge slimy green mess”) makes him seem like a real person. It’s a way for the NPC to interact with other NPCs (bringing up stories about them) without the other NPCs actually being there.
A patron or king can also show his inner self by how he acts to other NPCS. A king who is kind and solicitous with the player characters and then treats his own servants with haughty derision will be showing his true colors. This is a good way to foreshadow a betrayal. Likewise, a king that treats his servants with respect will be seen as a good force in the world and the players will be more willing to accept any task he has to offer; the players will bring their own likes and dislikes to the table and you can take advantage of that.
When defining an NPC there are 3 aspects that should be included; basically 3 lines of text to describe the NPC.
How the NPC interacts with the PCs.
How the NPC acts by himself.
How the NPC interacts with other NPCs.
Having NPCs interact with other NPCs is a way to breathe life into not only that NPC, but also your world as a whole. Every time one of your NPCs becomes more “real” the world becomes more “real” as well.