Learn Hardcore Japanese: kami wo tukue no ue ni oite kudasai.

In Japanese, を marks the direct object of a verb—in this case, 紙 (the thing you’re placing). に marks the destination or location where something is placed—here, 机の上 (on top of the desk). Together they tell you “put the paper (を) on the desk (に).”

Can I change the word order to say 机の上に紙を置いてください?

Yes. Japanese has flexible word order, so 机の上に紙を置いてください is actually more common and sounds perfectly natural. You could even emphasize the paper by saying 紙を机の上に置いてください or the location by saying 机の上に置いてください紙を, though the last one is unusual in polite speech.

What does the particle の do in 机の上?

の connects two nouns in a possessive or descriptive way. 机の上 literally means “the top (上) of the desk (机).” It’s like the English “of” or the ’s construction (“desk’s top”).

Why is the verb 置く in the form 置いてください instead of the plain 置く?

置いてください is the て-form of 置く plus ください, which turns it into a polite request (“please put”). The plain form 置く would be dictionary form (“to put”) and doesn’t express a request.

What’s the nuance difference between and the plain imperative form?