Think of a supplication as sort of a prayer, a request for help from a deity. The word carries a sense of awe and adoration with it, suggesting something tentative, even servile, a respectful appeal to a higher power.
Although it is a noun, supplication comes from the Latin verb supplicare, which means "to plead humbly." While a supplication is often thought of as a religious prayer (it is used 60 times in the Bible), it can logically be applied to any situation in which you must entreat someone in power for help or a favor. Still, it carries with it an aura of religiosity and deep entreaty, and should not be used to describe just any simple request.