Final answer:
The Supreme Court case that ruled states are required to provide attorneys for defendants who cannot afford one is Gideon v. Wainwright, which has had a substantial impact on the U.S. legal system and the development of the public defender system.
Step-by-step explanation:
The case that ruled that states must provide attorneys at state expense for accused persons unable to procure their own legal defense is Gideon v. Wainwright, 372 U.S. 335 (1963). The Supreme Court's unanimous decision declared that the Sixth Amendment's guarantee of counsel is a fundamental right essential to a fair trial and that it is obligatory upon the states to provide counsel for defendants unable to afford legal representation themselves. This landmark case not only overturned the Betts v. Brady decision but also extended the ruling to apply whenever an accused person faced the possibility of imprisonment.
Clarence Earl Gideon, a poor drifter, had been denied a lawyer and was forced to represent himself in a Florida court for felony charges. The Supreme Court ruled that this denial violated his constitutional rights. As a result, the Gideon ruling has had a significant impact, contributing to the establishment of the public defender system, wherein the government provides legal representation for those who cannot afford it.