When something important, such as your reputation or money, rides on a particular person or thing, it will be won or achieved if that person or thing is successful.

The future of the company now rides on the new managing director. 公司的未来现在就指望新任经理了。

I have a lot of money riding on that horse (= I will win or lose a lot of money if that horse wins or loses the race). 我在那匹赛马上押了很多钱。