The biggest problem with the Articles of Confederation was that it gave too much powers to the individual states and did not enumerate enough powers to the federal government. The abuses of King George III and the British were very fresh on the minds of Americans, and they were reluctant to give too much power to the national government out of fear that it might infringe upon their rights in the similar fashion.
Along those lines, the second problem was that the national government did not have the power to tax, which left the nation dependent upon the states sending revenue to the federal government. America was quickly lost financial stability under this plan, and the sale of land was the chief reason their were able to make ends meet.
The final problem was the ability of the Articles to make laws but the inability of the national government to enforce them. Again, that power was given to the states. This impossible situation was exposed by Shays's Rebellion. The states were unable to quell the resistance, but the national government didn't have the authority to intervene resulting in an unacceptable period of lawlessness. This situation was perhaps the greatest catalyst in causing Americans to look at their situation and realize that a change was needed to their government if America was going to be able to prosper in the future.
The main problem of the Articles of Confederation was that they did not give the federal government the power to tax. Keep in mind that the American Revolution was fought against a central authority who taxed the Americans without their consent--the Americans were loath to create the same thing for themselves. The newly minted United States of America owed debts to merchants in Europe and its own citizens, but states disagreed on whose responsibility it was to pay these debts.
Another problem was how easy it was for a state to block legislation. One example would be when Rhode Island, America's smallest state, vetoed a tariff that would have helped pay the American debt. The tariff failed and the government was left scrambling for answers. The Articles required 9 out of 13 states to agree in order to pass legislation.
While there are other problems, the final problem I'll talk about is the issue with Western lands. The United States gained control over land that went to the Mississippi River after the Treaty of Paris (1783). Many states claimed land that stretched all the way to this new boundary, and states had conflicting claims over this land. There needed to be a way to allocate this land fairly so that the citizens of these areas would feel as though they were an American citizen as much as someone from the original thirteen colonies.