These words refer to aversion or contempt towards anything one finds objectionable, distasteful or offensive.
Loathe is extensively used for simple distaste. It also focuses on aversion or dislike.
Ex – She loathed men who had mustaches or beards.
Despise usually indicates finding something offensive or morally objectionable.
Ex – He despised orthodox methods of punishing children.
Abhor is strongest among these words for suggesting outright hatred.
Ex – He abhorred men who had no moral principles.