The main issue of the Webster-Hayne Debate was the nature of the country that had been created by the Constitution. The debate can be seen as a precursor to the debate that became important at the time of the Nullification Crisis that came soon after it.
The question that Webster and Hayne were debating was what kind of country the Constitution had created. They had differing views of how closely the states were tied to one another by that document. Hayne believed that the Constitution was a contract made by the various states. He argued that the states were practically independent countries that had come together and made a contract with one another. By that reasoning, any state could back out of the contract if it felt that was a good idea. By contrast, Webster believed that the Constitution was created as a contract between all of the individual people of the United States. Because of this, no state could break the contract. Only the people as a group could do that.
This was a debate that would eventually help to bring the Civil War about. The South believed that the states had the right to secede while the North believed, with Webster, that they did not.