NPCs are the Non-Playable-Characters that exist in the world of Skyrim. Players can interact with NPCs through dialogue, or other means such as pickpocketing them or initiating combat with them. Talking to an NPC will start a conversation and dialogue options will be provided for the player to choose from. Through NPC interaction players can gain a follower, trade with them, get training from them and even become married to them. See below for the various types of NPCs.
NPCs have a hidden stat called Disposition, a measure of friendliness with the player. A low disposition can present negative dialogue and effects while positive dispositions can give you discounts at vendors, quests and even unlock the marriage option. Completing quests for them will help improve disposition.
Most NPCs can be killed by the player, some of them will eventually be replaced by a new random NPC but the majority will not be replaced. When dead you can loot their bodies for items and doing so is not considered a crime. Essential NPCs cannot be killed, even by the player, due to the importance of their role in the world and instead will go down on their knees and recover after some time. They may lose this status at some point through a quest or have scripted deaths as part of their quests. See the Essential NPC page for more information.
There are console commands that players can use that manipulate NPCs. This can be used to resurrect them back to life. See the console commands page for how this works.