The Federalists wanted a two-level government consisting of a strong, tightly-knit, centralized national government along with smaller, non-sovereign political units governed according to a national constitution. These small political units would be able to make laws on their own. Today, the United States has this kind of system with the federal government and state governments.
A main advantage of federalism is that it can help prevent the government in general from becoming too powerful since there is a dispersal of power from the national government to the states. Another benefit is that people can have a closer connection to the government since they can participate at the local and state levels rather than just at the national level.
One main drawback of this type of government, however, is that state and federal laws can often conflict. For example, gambling can be legal in one state but not in a neighboring state, and this can cause conflict among states.