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English Translation

To play a decisive or critical role

Breaking it down

1. 举: lift
2. 足: feet
3. 轻重: lit. light and heavy, here: severity of the case, whether something is slight or serious

Source: Google Image Search

Explanation in Chinese

一挪动脚；就会影响两边的分量。原指一个实力强的人处于两方之间；只要稍微偏向一方；就会打破均势。 比喻地位极其重要；足以左右全局。 语出 《后汉书·窦融传》：“方蜀汉相攻；权在将军；举足左右；便有轻重。”

Source: baike.baidu.com

Explanation in English

As soon as you move your foot, it will affect the weight on both sides. It originally meant that when a strong person stands between two sides, if he leans only a little bit on one side, then the balance of power will be broken. The metaphor means that the position is extremely important.
It comes from the "History of Eastern Han", in it says, the Fang kingdom and the Shu Han kingdom attacked each other, the power lies with the generals, moving the foot left or right a little bit will affect the severity on both sides.

Translation by Celine Li

Usage

李老师在我校是举足轻重的人物。
他意识到自己处在举足轻重的地位,所以下决心要与国家共存亡。