An ʿāmil (governor) is an element that causes or determines a grammatical case in a related word. The word governed is called the maʿmūl.
جَاءَنِي زَيْدٌ
Translation: Zayd came to me.
The verb «جَاءَ» governs its subject in the nominative.
رَأَيْتُ زَيْدًا
Translation: I saw Zayd.
The verb «رَأَيْتُ» governs its direct object in the accusative.
مَرَرْتُ بِزَيْدٍ
Translation: I passed by Zayd.
The preposition «بِـ» governs the noun in the genitive.