I don't own noise-cancelling headphones, but I have normal headphones and a macbook.
Is there an app I can get that will play the noise-cancelling noises through my headphones by using my macbook's microphone?
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There's no way to make a noise-cancelling program over speakers because the computer has no way of knowing distance from your ears to the speakers.
Noise-cancelling headphones scan for incoming sound and produce the opposite (canceling) wave. "Noise-cancelling headphones" are a misnomer - they counter incoming sound instead of blocking it.
If the noise-cancelling came from speakers, the computer would need to know where your ears are in real-time and that's just infeasible.