First of all, we must note that the US did not have a federal system right after the Declaration of Independence. Under its first constitution (the Articles of Confederation) the US had a confederal system. It was only after the ratification of the Constitution that the US came to have a federal system.
A federal system is one in which both the national government and the state governments have defined powers given to them by a constitution. The national government cannot take powers away from the states nor can the states take powers away from the national government. The Constitution laid out which powers the national government had and which powers the states had.
This is in contrast to what the US had right after the Declaration of Independence. Then, the US had a confederal system in which the states got to have much more power than the national government.