Both Barron v. Baltimore and Dred Scott v. Sandford deal with different clauses that appear at the end of the Fifth Amendment: "No person shall... be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."
In Barron v. Baltimore, construction by the city of Baltimore changed the flow of water in the harbor. Barron's wharf became useless, because the water around it was too shallow. Barron sued, claiming that the city's construction had deprived him of the use of his wharf, which violated the final clause of the Fifth Amendment: "Nor shall private property be taken for public use, without just compensation."
The Supreme Court, however, found that this clause of the Fifth Amendment doesn't apply to the states. It only applies to the federal government. Because the city of Baltimore is not a federal government entity, said the court, it couldn't be required to compensate people for taking their property.
By holding that the takings clause didn't apply to Baltimore, the Court created, in effect, a two-tiered system in which the federal government was required to abide by the Bill of Rights but the state governments were not.
In Dred Scott v. Sandford, Dred Scott began his life as a slave in Missouri. After living in Illinois, a free state, for ten years, Scott sued Sandford (his owner) in Missouri, claiming that the time he spent in Illinois had freed him.
The Court held that Dred Scott lacked standing to sue because his ancestors had been imported into the US as slaves. Ordinarily, a court would dismiss upon finding that one of the parties lacked standing; in Dred Scott, however, the Court went on anyway, holding that Illinois's law banning slavery could not free a slave, because doing so would violate the Fifth Amendment's prohibition on depriving individuals of property "without due process of law."
While the Court doesn't mention Barron in the Dred Scott decision, Dred Scott did contradict Barron somewhat. In Barron, the Court had held that the Fifth Amendment takings clause didn't apply to the states; in Dred Scott, the Court held that the Fifth Amendment due process clause did apply to the states.
Suddenly, the state and federal governments shared a responsibility to abide by the Fifth Amendment's due-process clause when dealing with citizens. While Barron didn't address due process, its holding implied that the states weren't responsible for abiding by the due-process clause. Dred Scott changed that relationship by holding that the due-process clause did apply to the states.