I was playing Baratheon and my opponent Targaryen.

I had Robert Baratheon in play with the "Lightbringer" attachment. Robert has 3 strength, and lightbringer is an attachment that can be kneeled to save the character from being killed.

My opponent played the 'threat from the north' plot card, which gives everyone -1 strength and says to discard characters who have a strength of 0. The plot card doesn't say, but presumably this effect persists for the duration of the round (i.e. from the plot phase through to the tax phase).

At the beginning of the challenge phase, she brought "Dragon Skull" out of shadows and attached it to Robert. The text of that card gives -2 strength to the attached character. It also says that the character is killed if its strength is 0.

So the question is which card takes precedence? Do I discard Robert Baratheon, or kill him?

While the -1 strength from TFTN is a constant effect, the "discard if 0" is a passive. The "kill if 0" on Dragon skull is a passive.

When two passives conflict, the order they are resolved is determined by the first player. In this case, whether Rob dies or is discarded is determined by the first player. Note that if he were under the protection of Power of Blood, he wouldn't die (but he would still be discarded, regardles sof which order the first player chose to resovle the two passives).

Lightbringer doesn't factor into this at all, because it's unable to save a character with a terminal state.

Edit: Effects with no duration or point of initiation, like Threat from the North, take effect while the card is revealed. The same is true for Fortified POsition, etc etc - they last from when they are revealed up until a new plot is revealed to replace them, including the player action window *before* plots are revealed the next phase.

Relevant section of FAQ is 2.5 - Simultaneous but Conflicing Entry into the Moribund State

Now, let's replace Robert with Ser Davos Seaworth (Core Set). His save is valid against Dragon Skull's effect (provided you don't pay 1 gold) because it will return him to your hand (where he will be out of reach. On the other hand, Threat from the North says "cannot be saved", so not even Davos can get himself out of this. Now the order of passive effects matters very much: if Threat from the North is resolved first, Davos cannot be saved and becomes moribund:discard, then Dragon Skull fails; if Dragon Skull is resolved first, Davos can be saved and becomes moribund:hand, then Threat from the North fails. Now I'll let you decide which scenario each player will prefer.