Responsibility is an internal game attribute between 0 and 100 for NPCs that determines the NPC's lawfulness. If an NPC has a responsibility of 30 or less, they will steal items that they need to complete actions. For example, if set to eat food, they will steal food (Ongar the World-Weary can be seen to do this regularly in Olav's Tap and Tack in Bruma).
Responsibility also affects an NPC's disposition toward you, depending on your infamy. NPCs with low responsibility will like you if your infamy is high. Similarly, NPCs with high responsibility will dislike you if you have high infamy. All NPCs, regardless of their responsibility, will like you more as your fame increases.
Merchants with responsibility below 30 will purchase stolen items. All the Thieves Guild fences have a responsibility that low, as do the merchants added by the Thieves Den Official Plug-in. Additionally, Raven Biter from Bliss in the Shivering Isles, Manheim Maulhand from the Inn of Ill Omen and the Dark Brotherhood Murderer from the Dark Brotherhood fall into this category.
If an NPC's responsibility is 100, they will immediately report crimes, giving you a bounty whether or not a guard is present. Note that performing a one-hit-kill sneak attack on somebody with a responsibility of 100—even without witnesses—will give you a bounty. Those with low responsibility, such as members of the Thieves Guild, will not report you, even if they see you commit a crime. Instead they will simply comment on the crime being committed.
An NPC with a responsibility from 90 to 30 will not report crimes as long as their disposition doesn't drop too low and will allow some of their items to be taken, after that they will report crimes, for example, a 50 responsibility NPC will not report crimes if their disposition is above 50.
Reaction dialogue with 30 or less responsibility (No bounty):
"Hah! It has been too long since I've seen a good brawl!"
"Woh! I hope you know what you're doing!"
"Fight! Fight! Fight!"
"Hey, what do you do that for? You're some kind of maniac?"
"Hey, what did I ever do to you?"
"Hey take whatever you want, it's none of my business."
"What? I didn't see a thing."
"Hey, you don't see me stealing your things!" (NPC owning the item will say this if witnesses the crime)
"You can take that one, but let's not make it a habit!" (NPC owning the item will say this if witnesses the crime)
"Hey, keep your thieving paws to yourself."
"Pfft! Take it! it's worthless to me anyway!"
"Still here? You've got nerve, I'll give you that!"
"Why don't you just go? We don't want any trouble."
"Stay! Go! Why do I care..."
"By the gods! How could you do such a thing?"
"Humpf! Better him than me!"
In general, an NPC's responsibility is a fixed value (shown on each NPC's page); there are only a few cases (Audens Avidius, the vampire hunters in Information at a Price) where a game script alters an NPC's responsibility.