There have always been political parties, they just were not always defined as political parties. Before the early 19th century, they were called faction. George Washington deplored what he called faction, and believed for the republic to truly offer democratic rule and justice, men of good will must avoid faction and focus only on the commonweal. The terms liberal and conservative as political labels, for example, only first appeared in the press in the early 1830s in England at the time of the first Reform Act. Thomas Jefferson is given credit by the Democrats as the founder of their party. That's not true. Arriving back in the United States from France in 1789, after the constitution had been ratified, he was disgusted to learn that it had been ratified and even though he took the post of Secretary of State in Washington's cabinet, he worked behind the scenes to interfere with policies he opposed, and especially made a failed attempt to sabotage Alexander Hamilton's financial measures as Secretary of the Treasury. Those who supported the ratification of the constitution were known as the Federalists, and if referred to at all, those who opposed the constitution were called anti-Federalists. Jefferson was enough of a politician to know that a faction--a political party--cannot succeed if it is defined only by what it opposes. So he founded the Democratic-Republican Party along with James Madison in the period 1791-93, Washington was so disgusted with the game Jefferson had played while a member of his administration that he never spoke to him again. Hamilton's Federalists were well enough organized to succeed in getting John Adams elected in 1796, but with men like Jefferson and Madison fanning resentments, the Republicans (as the Democratic-Republicans were usually called) were a steam roller, and the Federalists were doomed. They just didn't know it right away.
Jefferson won the 1800 election, and was followed after two terms by James Madison, who was succeeded by James Monroe. The Federalists were able to send people to Congress, but were virtually without national influence by the end of Madison's first term. The United States had become a one party nation. It was Andrew Jackson who founded the Democratic Party, and in the process created the first modern political party in the history of the United States. In 1824, Jackson and several other members of the Democratic-Republicans stood for President. The final field had four members--Jackson, John Quincy Adams, William Crawford and Henry Clay (that is the order in which they finished in the popular vote and the Electoral College). As none of them had gotten 50% of the popular or the electoral vote, the election was thrown into the House. By the constitution, only the top three were to be voted on. Clay detested Jackson (he wrote to a friend that he didn't think that killing 2500 Englishmen at New Orleans qualified Jackson for the office) and Crawford was known to be a sick, old man, so he threw his support and considerable influence to Adams. Jackson despised Clay as much as Clay despised him, but he was a man who would get even as well as getting angry.
Jackson organized his new Democratic Party from the ground up. Men who had served with him in the Creek War of 1813, at New Orleans and during the invasion of Florida now became his precinct committeemen and county chairmen. The party was organized in the same manner in other states, and successfully because Jackson was already seen as a champion of the "common man." He trounced Adams in the 1828 election. In the 1832 election, on the wave of Jackson's popularity, the Democrats took over most state houses. It was at that time that states, with the constitutional authority to certify their own elections, began passing the winner-take-all legislation for Electors which make the Electoral College so pernicious today.
Clay and others organized the Whig Party, but they really didn't fully understand what Jackson had done. The Whigs were rarely a power in Congress, and only elected two Presidents--William Henry Harrison and Zachary Taylor, both of whom died in office. The early success of Harrison's election in 1840 was not a harbinger of political power, and the Democrats dominated American politics until 1860. Only one other party in American history was organized from the ground (or grassroots) up in the Jacksonian manner, and that was the Republican Party of the 1850s, from which the modern Republican Party is descended.
Make no mistake, though, there has been faction, political parties in all but name, for more than 2000 years. It can be seen in the History of the Peloponnesian War
written by Thucydides 2500 years ago; it can be seen in the history of the Roman republic. It can be seen in the history of the French revolution. When the royal family were brought back to Paris in October 1789, the National Assembly set up in the indoor riding arena at the Tuileries Palace. From the President's table, the constitutional monarchists were on his right and the anti-monarchical republicans on his left--which is why we call conservatives the right and liberals the left. There have always been and there will always be political parties.