Modus tollens is fun because it is often applied incorrectly in informal arguments to come to the wrong conclusion. Here's an example.
The other day I saw a tweet which read:
"A fear of weapons is a sign of retarded sexual and emotional maturity." - Sigmund Freud, General Introduction to Psychoanalysis (1952)
Simplifying that a little (and assuming we agree with Freud) it can be written "If a person is fearful of weapons, then that person is sexually and emotionally immature". If we write "fearful of weapons" as F and "sexually and emotionally immature" as I, this statement can be rewritten F => I (the => is read as implies and make the entire statement read as F implies I or If F, then I).
So if we come across someone who satisfies F (i.e. they fear weapons) then we know that I applies (i.e. they are sexually and emotionally immature).
Modus tollens tells us that if we come across someone who does not satisfy I (i.e. that person is sexually and emotionally mature) then we know that they do not satisfy F (i.e. they do not fear weapons). Symbolically that would be written ~I => ~F.
Modus tollens tells you that if the "then" side of an "if, then" is false then the "if" side must be also (this has to be the case because the "if, then" forces the "then" side to be true when the "if" side is true).
The common fallacy related to modus tollens is to think that ~F => ~I follows from F => I. That is, given Freud's statement some people will believe that the statement "People who are not fearful of weapons are sexually and emotionally mature". This is called the Denying the antecedent fallacy.
Next time you come across someone who doesn't fear weapons you'll know that Freud tells us nothing about their sexual or emotional maturity.