How can I use 其 properly? I wonder how I should use 其. To express possessive pronoun, you can use 他的，它的，他们的，etc.

So when should I use 其? Can it function as just a replacement for these possessive pronouns?

(And I just found that it is also used as a subjective pronoun...)
其, in ancient Chinese, is used as a personal pronoun: he, she, it, his, her, its. nowadays, in Chinese, we partially inherited its usage. That means it will only make sense in some fixing words, phrases and sentences. besides examples by haksayng, there are more (which are commonly used in my quotidian world): 以其人之道，还治其人之身 （meaning "fight fire with fire" or "tit for tat"）, 有其父必有其子 （meaning "like father, like son"）, etc...