The sentence that correctly uses a nonrestrictive clause is:
"The dog—who is black and brown—jumped over the fence."
In this sentence, the clause "who is black and brown" provides additional information about the dog but is not necessary to understand which dog is being referred to. Nonrestrictive clauses are usually set off by commas (or dashes, as in this case) to indicate that they are not essential to the main meaning of the sentence.