What is the correct usage: Human Resource Department or Human Resources Deparment?

The correct name is the Department of Human Resources.

Unfortunately, "it depends" is the accurate answer. "Human resources" is a kind of administrative service, so when not used in a proper name, it shouldn't be capitalized. (Example: "She studied human resources at the University of Whatever.") When used as part of a proper name, as in the name of a department within a business, you should conform to whatever name that business uses (the Department of Human Resources at Such-and-Such, Inc., or the Human Resource Department at Whatever, Co., or the Human Resources Department at Yada-Yada).

That said, I would tend to go with Department of Human Resources to avoid risk of error. If you must put the term before the noun, I'd say "Human Resource Department," since "human resource" becomes an adjective, and adjectives are never plural. Keep in mind, though, that "human resources" is in the dictionary as a noun; "human resource" is not; so the only 100% correct way would be "Department of Human Resources."

The reason you see inconsistency, though, is because there is no clear answer. There is no definitive answer regarding rules of style; it's about clarity and consistency. The important thing is to conform to the company's preferred style.