In English, the object of a preposition is the noun or pronoun phrase that follows a preposition. Therefore, in the sentence "The king of Italy lost the handkerchief in a storm," "Italy" is the object of the preposition "of," and "a storm" is the object of "in."
When the object of a preposition is a pronoun, writers must be sure to use the object case of the pronoun. For example, "Tom saw a storm over them" is correct because it uses the object pronoun "them" after the preposition "over." "Tom saw a storm over they" is wrong, because "they" is the subject form of the pronoun. This is also true with "who," which is a subject pronoun, and "whom," which is an object pronoun. Therefore, "On whom was the picture tattooed" is correct, while "On who was the picture tattooed" is wrong.
Sometimes, the object of the preposition is very long, and sometimes it is very short. Here is an example of the former: "The king of the lands filled with gluttonous cannibals who frighten everyone visited our house today." The object of the preposition is the entire noun phrase "the lands filled with gluttonous cannibals who frighten everyone." This prepositional object contains another prepositional phrase ("with gluttonous cannibals who frighten everyone") that itself contains an adjective clause. However, the whole noun phrase is the object. In this sentence, however, the noun phrase that functions as a prepositional phrase is very short: "The man in China was brave." The noun phrase "China" is the only word in the object of the preposition "in."