"We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America."  --Preamble to the United States Constitution

The founding fathers of this country faced a crisis in the 1780's. The economic problems of taxation and debt could not be cured by the current system of government under the Articles of Confederation. The reason the Articles were inadequate stemmed from a lack of federal leadership and the legislative inefficiencies therein. Further inflaming the fears of politicians was a rebellion of farmers in Massachusetts, called the Shay's Rebellion. For these reasons, the forefathers headed to Philadelphia for the purpose of revising the law that would govern the land.

The true purpose that guided the framers of the Constitution is found in the Preamble and all point to significant problems with the Articles. It is useful to analyze the various components of the Preamble to understand the purpose of the Constitutional Convention.

 .....in order to form a more perfect union

The old system of government was a confederation by design.  That meant that each state was more powerful than the national government.  This created the opposite of unity or union.  For that reason, citizens did not identify themselves as Americans as much as they would as Virginians, as an example.

.....provide for the common defense, promote the general welfare..

The Articles of Confederation hampered a national standing army for defense and security.  Because Congress lacked the ability to levy taxes, it was difficult for the federal government to support a defense system.  The lack of clout at the federal level in terms of regulating commerce also made it difficult to promote strong economic development and, therefore, a sense of general welfare among the citizens.  

 ...secure the blessings of liberty to ourselves and our posterity...

Despite the revisions made in the U.S. Constitution that gave the federal government more power and authority, the Framers were still concerned about liberty and freedom.  They hoped to strike a balance between the weaknesses of the federal government under the Articles and the tyranny of the monarch government system that they had just broken ties with.  The result of the Constitution, at its completion, was a system of government with three branches to separate responsibilities and powers that were reserved to the states.

According to James Madison's own diary from 1787, the purpose was to convene and revisit the Articles of the Confederation, which constituted the first "rule book", so to speak, drafted by the forefathers of the nation in order to decide how to rule it, and what regulations would be put together to maintain order.

In his diary, Madison refers to the Friday, May 14 meeting the following way:

Monday May 14th 1787 was the day fixed for the meeting of the deputies in Convention for revising the federal system of Government. On that day a small number only had assembled. Seven States were not convened till, Friday 25 of May.

That federal system of government that Madison talks about are the Articles of the Confederation mentioned previously. As the writing says, it was not until the 25th that everyone actually came together. This tells you that the process of bringing the country together was no simple task. Everyone had an idea of the type of government that would be best for the country, but the entire concept of the "United States" that we have in the 21st century was nonexistent in the 18th century, at least in practice, and certainly not in this part of the world. Therefore, while hard and heavy, the task was finally accomplished and, like the previous poster correctly pointed, the result was to reinforce the central power of the federal government.

The official purpose of the Constitutional Convention that met in Philadelphia beginning on May 25, 1787 was to amend the Articles of Confederation.  It had, by that time, become clear that the Articles of Confederation were not a good enough constitution for the new nation.  The convention was called to improve on the Articles.  However, the convention was not called to do what it ended up doing.  What it ended up doing was creating an entirely new constitution.  This new constitution (the one we have today) gave the federal government much more power than it had had under the Articles of Confederation.