I would say that the Federalists fell from power because their policies got to be unpopular among the people. there were quite a few policies of theirs that contributed to them losing power.
First, people were unhappy about the Alien and Sedition Acts. Those laws made it seem as if the Federalists were against freedom.
Second, people were unhappy about the power of the federal government. They worried that the Federalists wanted a monarchy and they pointed to things like the Whiskey Rebellion as proof that this was the case.
Finally, people were unhappy because they felt that Jay's Treaty had given up too much to the British.
Please keep in mind that the Republican party to which Talon06 was referring is NOT today's Republican Party, which was founded in 1854.
Thomas Jefferson's Republican Party eventually evolved, in one form or another, into today's Democratic Party. It's original name was Democratic-Republican Party. It was the presidency of Andrew Jackson that a splinter group of the Jefferson's Republican Party started to call themselves the Democratic Party.
So, it could be argued that today's Democratic Party originated with Andrew Jackson or grew out of Jefferson's Republican Party.
In addition to what has already been said, the Federalists were dominated by business men. These rich business men did not trust the government and thought that elite society should be in charge. The Federalists favored national power over state power.
John Adams was the first and only Federalist president. Thomas Jefferson was the first Republican president. At that time, they favored state power and did not trust "big business" and the banks.
The Federalists eventually lost power as the Republicans became more popular with the "average man" and not the elite of society.