Final answer:
The case related to the states' rights versus national government power struggle is Dred Scott v. Sandford (1857), which was followed by a shift in power towards the national government after the Civil War, further solidified by McCulloch v. Maryland and Gibbons v. Ogden cases.
Step-by-step explanation:
The case that was related to the issue of states' rights versus the power of the national government is Dred Scott v. Sandford (1857). This Supreme Court decision ruled that the national government could not ban slavery in the territories, which delighted southern states but incited protests in the northern states. Furthermore, the Civil War elevated the debate on states' rights versus national authority, leading to the state and national power balance shift towards the national government after the Union's victory. This shift was captured through the enforcement of the Fourteenth and Fifteenth Amendments among conditions for the southern states' readmittance into the Union. Additionally, McCulloch v. Maryland (1819) and Gibbons v. Ogden (1824) are earlier cases where the Supreme Court affirmed the principle of national supremacy and maintained federal authority over states' claims to power.