1. It couldn't raise a national army. The defense of the country was dependent upon the state militias. Thus, if a problem arose in a state which was detremental to the national interest, the national government couldn't do a thing to enforce it. Not to mention, this led to a weak defense system of the U.S.
2. Congress could not collect taxes or issue national currency. As a result, there was no national budget, and the value of state currencies was unknown. For instance, if a chicken was worth $3 in Georgia currency, it might be worth $15 in South Carolina currency. As a result, people often got their money swindled from them. This led to instances such as Shay's Rebellion in which Daniel Shay, a farmer, led a revolt because he had lost so much money.
3. The Articles of Confederation allowed states to make treaties on their own, and the national government couldn't stop them. While the national government (Congress) could make a treaty with a nation, each state could make its own separate treaty. As a result, if Georgia wanted to make a treaty with England, and South Carolina wanted a treaty with France, this could lead to major problems between the two states.