The object of a preposition is the noun or pronoun that a preposition is relating to the rest of the sentence. The object of the preposition will usually be at the end of its containing prepositional phrase and is often separated from the preposition itself by one or more modifiers.
Andrew works in the kitchen.
The preposition in starts the prepositional phrase in the kitchen. The noun that in is relating to the sentence is kitchen. This phrase is functioning as an adverb to modify the verb works.
Our cousin from Norway missed his flight.
The preposition from starts the prepositional phrase from Norway. Norway is the object of the preposition being related to the rest of the sentence. This phrase functions as an adjective to describe the noun cousin.
The supervisor gave this responsibility to me.
The object of the preposition to in this sentence is the pronoun me. This phrase is functioning as an adverb to modify the verb gave.