Amour, liaison, intrigue and affair all denote an instance of illicit sexual relationship.
Amour is particularly applied to the illicit attachment of prominent persons; it stresses passion as the motivating force and often connotes transience.
Liaison implies duration but not necessarily permanence in the attachment; it is commonly used to designate the relation between a man and his mistress.
Intrigue emphasizes the clandestine element in the relation and is often closer to amour than to liaison in its other implications.
Affair is the least specific term and often suggests equivocal rather than definitely illicit relations or may be used without imputation of impropriety.