Literally, it comes from the word for “bird”, πουλί, which is originally from Latin. (There is a minority opinion that it comes from πώλος, “foal”.) The word has ended up being used as a diminutive, and Greek patronymic surnames are formed as diminutives. Diminutives in Greek vary by region, which means that you can tell someone’s region of ancestry from their surname.
–opoulos is the default suffix and diminutive of the Peloponnese. Exceptionally, while Standard Modern Greek is based on the Peloponnese, –opoulos is not the default diminutive: –aki is (used in Central Greece). –opoulo is used more narrowly in Standard Modern Greek, though, to denote young animals (γουρουνόπουλο, “piglet”), and children of nationalities (εγκλεζόπουλο, “English kid”).
So Papadopoulos means “son of a priest”, as Konstantinos Konstantinides says, but originally παπαδόπουλο means “priestling”.