The distinction is used regularly in analytical metaphysics and in philosophy of language.
There are two possible interpretations of the sentence "Peter believes someone is out to get him": On the de dicto interpretation, 'someone' is unspecific and Peter suffers a general paranoia; he believes that it is true that a person is out to get him, but does not necessarily have any beliefs about who this person may be.
On the de re interpretation, 'someone' is specific, picking out some particular individual.
In the context of thought, the distinction helps us explain how people can hold seemingly self-contradictory beliefs.
[4] Say Lois Lane believes Clark Kent is weaker than Superman.
Since Clark Kent is Superman, taken de re, Lois's belief is untenable; the names 'Clark Kent' and 'Superman' pick out an individual in the world, and a person (or super-person) cannot be stronger than himself.
Understood de dicto, however, this may be a perfectly reasonable belief, since Lois is not aware that Clark and Superman are one and the same.
Consider the sentence "Jana wants to marry the tallest man in Fulsom County".
One interpretation is that Jana wants to marry the tallest man in Fulsom County, whoever he might be.
On this interpretation, what the statement tells us is that Jana has a certain unspecific desire; what she desires is for Jana is marrying the tallest man in Fulsom County to be true.
Another way to understand the distinction is to ask what Jana would want if a nine-foot-tall immigrant moved to Fulsom county.
If she continued to want to marry the same man – and perceived this as representing no change in her desires – then she could be taken to have meant the original statement in a de re sense.
If she no longer wanted to marry that man but instead wanted to marry the new tallest man in Fulsom County, and saw this as a continuation of her earlier desire, then she meant the original statement in a de dicto sense.
Take the sentence "The number of chemical elements is necessarily greater than 100".
In de dicto claims, any existential quantifiers are within the scope of the modal operator, whereas in de re claims the modal operator falls within the scope of the existential quantifier.