Carol laid the book on its side on the desk.
A. its
B. is wrong because (its) is already the possessive form of the pronoun, the apostrophe does not belong.
C. is wrong because (it) is not a possessive pronoun.
D. is wrong because (it's) is a contraction for (it is) and you wouldn't say - Carol laid the book on (it is) side of the desk.
Hope this helps. :)