First, we have to realize that it is not objectively true to say that the Articles of Confederation were a failure. This is more of an opinion than a fact. After all, the Articles were in force as the US won the Revolutionary War and kept the country together for another eight years after that. This at least makes it possible to say they were a success.
If we do say the Articles were a failure, it was because they did not give the federal government enough power. The Articles were written at a time when the Americans were trying to break away from England, which had a strong central government. Therefore, the Articles created a very weak central government in response. The government was not able to tax the people. It was not able to require the states to do anything. It was not even able to regulate trade between the states. It did not have a president and it did not have a judicial branch.
Because of these things, the US had some problems. One of the problems was that states came into conflict with one another. They had trade wars with one another, harming the overall economy. The inability to tax made it impossible for the US to create any sort of a strong military. These sorts of problems led some in the United States to feel that the Articles had failed and to call for a new constitution.
The Articles of Confederation could be considered a failure primarily because it lacked a centralization of power. To contrast with the monarchy of Britain, the States wanted to avoid having too much federal power, but rather engage in a league of friendship where each State could maintain its own sovereignty. In the end, this lack of federal power would hurt them quite a bit.
Looking at the role of the federal government, I would say that its two biggest limitations were that it couldn't levy taxes and it couldn't have a federal military (though it could declare war). Since it couldn't levy taxes, they had a difficult time gathering up revenue from the States, leading to a nation government that was financially challenged. As well, no federal military led to issues in instances like Shays' Rebellion, where no national forces could be used to stop the revolts. Overall, the Articles of Confederation made the federal power look incredibly weak. There wasn't even an executive or a judicial court, but just a Congress. They had very little significant powers or abilities and as a result, weakened the concept of united States.