a counter example to a syllogism is a syllogism of the same form as the original, but with obviously true premises and false conclusion, in order to show the original to be invalid.
key point #1
the validity of syllogisms depends solely on their form. Certain forms are always valid; other forms are always invalid.
do not assume a syllogism is valid if you cannot think of a counter example to it. You may just need to be more creative.
key point #2
when developing counterexamples, try starting with a false conclusion, then find a middle term which results in true premises.
summary (something i'll try to include from now on)
an invalid argument may be exposed through the use of a counterexample. a counterexample is a syllogism of the same form as the original argument, but with obviously true premises and an obviously false conclusion.