Forms
E	I
No S are P. Therefore, no non-P are non-S.	Some S are P. Therefore, some non-P are non-S.
Venn Diagrams
Similar Validating Forms
A	O
All P are Q. Therefore, all non-Q are non-P.	Some P are not Q. Therefore, some non-Q are not non-P.
Examples
No conservatives are liberals. Therefore, no non-liberals are non-conservatives.	Some dogs are pets. Therefore, some non-pets are non-dogs.
Counter-Examples
No dogs are cats. Therefore, no non-cats are non-dogs.	Some physical things are invisible. Therefore, some visible things are non-physical.

Exposition:

"Illicit contraposition" is not a scandal from the Reagan administration. Rather, "contraposition" refers to the process of switching the subject and predicate terms of a categorical proposition, and negating each. For instance, the contraposition of "all bats are mammals" is "all non-mammals are non-bats".

Contraposition is a validating form of immediate inference for A- and O-type categorical propositions. However, contraposition is a non-validating form of inference for E- or I-type propositions. Hence, contraposing an E- or I-type proposition is Illicit Contraposition.

Exposure:

The Venn Diagrams above show that the two forms of Illicit Contraposition are not validating. The diagram for the E-type proposition shows that no S is P―which is indicated by the blue shading of the overlap of the S and P circles―but it fails to show that there is nothing in the area outside both circles, which is the overlap of the non-S and non-P areas. A question mark in an area means that it might or might not be empty. Similarly, the diagram for the I-type proposition shows that some S is P, but fails to show that there is anything in the area outside both circles.

Moreover, the Counter-Examples above show that the forms are not validating, since each has an instance with a true premiss and a false conclusion. It's harder to find a Counter-Example to the contraposition of I propositions than E ones since, in order for the conclusion of such an I inference to be false, the two classes must jointly exhaust the universe but not be mutually exclusive. In the Counter-Example, the premiss is true because there are invisible physical things, such as atoms; but the conclusion is false because there are no visible things that are not physical.

Source:

Patrick J. Hurley, A Concise Introduction to Logic (Fifth Edition) (Wadsworth, 1994), p. 214.

Acknowledgment: Thanks to John Congdon for the "contra" joke in the Exposition.