mitochondria has it's own DNA. mtDNA is passed from mother to daughter and mother to son.
Here's Miss Rose with her red strand of mtDNA, or a piece of Mizuhiki Cord. She'll pass that mtDNA to all her children.
Her husband, Mr Brown has a wide blue strand of mtDNA that came from his mother, but he can't pass that on to his children.
Mr Brown also has some blue Mizuhiki Cord. That's the Y-DNA he inherited from his father. Miss Rose doesn't have any Y-DNA because that Y chromosome is what makes an individual a man. Mr Brown will pass his Y-DNA to his sons, but not his daughters.
Got that? Now let's look at a pictorial tree of the Paper Family. Son Brown marries Miss Vines and Daughter Brown marries Mr Black. Each of those couples has one son and one daughter. What Y-DNA and mtDNA will those grandchildren have?
Notice that Granddaughter Black has Miss Rose's mtDNA while Grandson Brown has Mr Brown's Y-DNA, as does son Brown.
Next time: the DNA testing options.