The measure of a single word is comparable to that of the nine tripod cauldrons (see Notes)
Refers to words that are influential or play a decisive role in something
Positive
老王是一言九鼎之人，说话绝对可信 | Lǎo Wáng shì yīyánjiǔdǐng zhī rén, shuōhuà juéduì kě xìn
Old Wang's words carry a lot of weight, and he is absolutely credible
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他是个说话一言九鼎的人，值得大家信任 | Tā shì gè shuōhuà yīyánjiǔdǐng de rén, zhídé dàjiā xìnrèn
He is someone who is greatly influential and deserves everyone's trust
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A 鼎 (dǐng) is an ancient cooking vessel, sort of like a cauldron. They were generally round and had three legs, though there are square four-legged varieties. In a similar vein, 九鼎 refers to a set of nine 鼎 that were said to have been cast by the mythical 夏禹 (Xià Yǔ), the ruler of the Xià Dynasty (2070 - 1600 BC). This set of 鼎 represented the nine major divisions of power at the time.
The background of 毛遂 (Máo Suì), who is mentioned in the below story, can be seen in the chengyu 毛遂自荐. I recommend reading that first so that this chengyu story makes more sense
《史记》(Shǐ Jì), or the famous work "Records of the Grand Historian" by 司马迁 (Sī Mǎ Qiān).
In year 290 BC, the kingdom of Zhào (赵国) found its capital city of 邯郸 (Hán Dān) surrounded by enemy Qín kingdom (秦国) troops. The Zhào king sent the lord 平原 (Píng Yuán) and tens of other men to ask the neighboring kingdom of Chǔ (楚国) for help. When they arrived, the king of Chǔ was hesitant to oblige, and negotiations temporarily stalled. At that point, 毛遂 (Máo Suì), one of the men who had traveled with 平原, approached the king without permission and presented an excellent case as to why the Chǔ should render assistance. The king was impressed by 毛遂's daring and logic, and he agreed to send help. Not long after, the combined armies of the Chǔ and the Zhào forced Qín to fall back.
Later, when speaking about 毛遂, 平原 said that his words were worth the nine 鼎.