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. Some people, such as Thieves Guild members, will not report you, even if they see you commit a crime - they simply comment on the crime being committed. For example, "I haven't seen a good brawl in a long time."
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.