The Federalists were a group of very powerful politicians who were in favor of limiting States' Rights and strengthening the role of central government. Alexander Hamilton was in many ways the leader of the party which basically lasted for about twenty years between 1792 and 1816-1820. Almost all of their support came from the North and the North East and New England in particular, essentially divided long the same lines as the country would be divided in the Civil War not too many years down the road.
They believed that a strong central government was key when it came to dealing with international politics as well as being able to mount a legitimate defense of the country in times of war, etc., particularly after the difficulty of raising tax revenues when there was no real federal power to do so and the funds were "requested" from the states.
In general, the Federalists were richer people who mostly lived in the New England area (but rich people in all states were more likely to be Federalists). This is of course not completely true, but it is generally true. These were the elites of the society, as can be seen in their views.
The reason why they supported a strong central government is that they were afraid of the influence of the masses. They felt that they masses were poor people who would vote in line with their material needs rather than with what was good for the country. They thought state governments were more likely to be dominated by the voting power of the masses. By contrast, they thought that a strong national government would be more insulated from this influence.
So what they wanted was a strong central government that would protect the few from the many. They looked to episodes such as Shays's Rebellion and said that democracy led to that sort of violence. They said it led to destructive economic policies such as stay laws. In general, they thought too much democracy led to bad laws that would destroy the country's economy. This is why they wanted a strong central government.