If a creature has an ability that is like a power. For example, an undead with fear ability. Are the character allowed to save using their Arcane Resistance Edge?
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No. Arcane Resistance only gives an armor bonus or Trait roll bonuses when "hit by damage-causing arcane powers, and […] when resisting opposed powers" (SWEX, p. 20).
Additionally, Arcane Resistance only works against the type of powers chosen when the Edge is bought. The choices available match the Arcane Backgrounds available in your game.
For Fear specifically, Arcane Resistance doesn't apply unless the fear is the effect of a Power. The Fear rules on page 100 of the Savage Worlds Explorer's Edition are independent of the Powers system and aren't affected by Arcane Resistance (although see the final paragraph below for a possible exception).
Even then, it's a judgement call on a case-by-case basis depending on the fiction. It can be easy to forget that Savage Worlds is a "fluff matters" system. If it's a "magic spell" (assuming that the PC with AR chose "magic" and not one of the other Resistances), then Arcane Resistance applies. If it's a natural effect merely being modelled as a Power, that doesn't qualify as a "magic spell" and Arcane Resistance doesn't apply.
Four cases to illuminate the difference:
However, it's worth emphasising again that Savage Worlds is a "fluff matters" system, so Arcane Resistance may also apply to non-Powers if the GM judges that the fluff warrants it. For example, if a character has Arcane Resistance (weird science) and is standing next to the Doom Engine when it explodes, the GM might rule that the explosion is partly arcane (weird science) and grant the Arcane Resistance benefits to avoid some of the explosive damage. Similarly, if the characters are travelling on the Plane of Magic, which causes periodic checks to see if the characters are altered by their exposure to raw Magic, then perhaps the character with Arcane Resistance (magic) gets the AR benefits on that check and/or suffers less significant alterations.
The key in all cases is the answer to the question, "Does the game fluff say this is a magic/pscionic/etc. effect?" If the answer is "Yes", then the matching Arcane Resistance will apply. If not, then it won't.