Federalism is a system in which there is a constitution that divides powers in a country between a national government and lower levels of governments. This is the sort of system that the United States has.
In the United States, the federal government and the state governments each have their own set of rights and powers. The federal government cannot simply tell the state governments what to do in all cases. For example, the federal government cannot tell a state to lower its income tax rates. At the same time, however, the state governments cannot tell the federal government what to do in all cases. For example, the state governments do not have the right to tell the federal government how to regulate commerce between the states.
Federalism is typically put into place in countries that are large and/or countries in which the population differs in important ways from region to region. The US Constitution imposed a federal system because the country was very big and because the states were very different from one another on issues such as slavery.