Pivotal stones are important because if they die, the enemy forces are joined up; if they live, the enemy remains divided and subject to separate attack. Not correctly distinguishing cutting stones from non-cutting stones is a typical beginner mistake.
Pivotal stones is a direct, character-by-character translation of the Japanese kaname-ishi, which as a compound also means keystone (of an arch).
Non-pivotal stones, junk stones, or kasu-ishi in Japanese, have a corresponding proverb: Do not run away with junk stones.
Here is an example from Go World 105, p. 15, Yu Bin vs. Cho U, 9th LG Cup, Game One. Discussing , the commentary says "Players proud of their fighting strength would probably attack with and in the diagram without even thinking. But if White cuts at , Black counters by pushing out at . If , captures the pivotal stones."
From The Magic of Go, #62. The game is from the 1999 European Fujitsu Finals. The commentary states: "If Black 67 is played at 1 in Variation 3, Black will have no real answer after white 6. If black 7, white 8 puts Black on the spot making miai between capturing the marked stones and the pivotal stones at 3 and 5."