Hamilton was seen as a person who favored more power obtained by the central government. This means he was in favor of things like a national bank, less state rights, and less of an ability for freed slaves to vote (because federalists were also democrats). Jefferson, on the contrary, was an anti-federalist. He feared that excess central power would infringe on the rights of the states, and so to avoid this, he did as much as he could to retain state rights. He also believed central government authority gave more power to the upper class (wealthy whites) than it did to the common man (poor whites, freed slaves).
The idea is known as the Republican form. The idea is that people democratically elect those who would represent them in a government, and those people would be held accountable for what they've done if they do something opposite to the will of the people. It's the only way that a democracy can work properly because others would be impossible with that many people.
There are several elements to this tradition. Freedom of speech, a democracy system where not just one person rule and makes decisions, and a jury system.
