(Gukou yama wo utsusu; “The Old Fool Moves Mountains”)
No matter how great your troubles or difficulties, hard work and perseverance will see you through. Your goals are realized through hard work and persistence. Rome was not built in a day.
In this case, 愚公 (gukou) is a proper noun. Not a name so much as a title, it effectively means “(The) Foolish Old Man.” As is often the case, modern Japanese might use a particle next, but it isn’t strictly necessary; instead we simply move on to the noun 山 (yama), “mountain,” with the object-marker particle を (wo). The mountain is taken as the object of the transitive verb 移す (utsusu), “to move (something)”; the verb is in sentence-final form, meaning that despite its brevity this kotowaza is in fact a complete sentence.
愚公 is not an actual word in Japanese. 愚考 and 愚行 (foolish ideas and foolish actions, respectively) both are, and have the same pronunciation, but their use in this saying would be incorrect.
This comes from an ancient and famous Chinese folk tale, attested since at least the 5th Century BCE, when it was recorded in the Liezi, a Daoist text.
One day, an old man decided that he needed to remove some mountains near where he lived, because they made it difficult to transport things. He began working on removing them by hand, with pick and basket. The people made fun of him for attempting an impossible task, to which he replied that if he, and his children, and their children and further descendants all worked tirelessly, then eventually the offending mountains would be cleared away. Shangdi heard of the old man and was so impressed with his grit that he ordered the mountains moved, so that the work was finished in a single day.
Apparently Mao Zedong was very taken with this story, and popularized as a call for collective action by the Chinese people. It is not nearly so well known in Japan, given that it must compete for head-space with the synonymous 雨垂れ石を穿つ.
(“Gukou yama wo utsusu to iu rei mo aru kara, watashi mo seikou suru made hirumu koto naku ganbaru tsumori da.”)
[“Following the example of the Old Fool Who Moved Mountains, I too shall work without faltering until I succeed.”]