其 + [noun]
使 / 令 / 让 / 要求 + 其 + [verb phrase]
其 points back to someone or something named earlier, in writing. Before a noun it means "its", "his", "her" or "their" – one word for all of them, and the sentence before tells the reader which.
其 already means "its", so it takes no 的. It often follows 及 "and": 公司及其员工 "the company and its staff".
After 使, 令, 让 or 要求, 其 stands for the person or thing that then does something: "make it…", "let them…", "ask him to…". 他 or 他们 would sound spoken in a formal sentence; 其 keeps the register.
其 can't start a sentence as its subject the way 他 or 它 does. In that position, use the ordinary pronoun or repeat the noun.
其他, 其中, 其实, 尤其 and 极其 contain 其 but are learned as whole words. 其中 has its own entry, as does the first pass at 其 at HSK 4.