Layman enthusiast here, I'll do my best in the absence of someone more qualified.

There were several points of high risk, but at one notable point late in the crisis a routine American military flight went off-course. The pilot got lost and somehow ended up flying over the Soviet Union.

Putting yourself in the shoes of the Soviets, imagine what a lone American plane flying into Russian airspace at the height of a nuclear weapons crisis must have looked like. When Robert McNamara (the Secretary of Defense) was informed about the flight I recall he blanched white and exclaimed "But this means war with the Soviet Union!" (Paraphrasing there.)

Of course it didn't, thankfully. The pilot of the wayward plane was allowed to land in the Soviet Union.

There was also a significant crisis point involving a Soviet nuclear submarine you could look up, but my memory of that is more sketchy and it would be a bad idea for me to attempt an full explanation.

Source: One Minute to Midnight, a popular history by Michael Dobbs.