In a country that employs a unitary government, governmental power is concentrated at the national level. A unitary government is sometimes referred to as a centralized government because power is concentrated in one place. The role of local governments is generally limited to implementing and enforcing the laws and policies enacted at the national level. Local governments don't make laws or public policies to any significant degree.
While this structure may seem odd if you are a citizen of the United States, most countries are organized under a unitary system of government. Familiar examples include the United Kingdom, France, Japan, China, Finland, Denmark, and Norway just to name a few. If you're a US citizen, it may help you to understand the unitary form of government by looking at how your state government is set up. Each state in the United States actually employs a unitary form of government; any power that counties, cities, or towns may have is only based upon what the state government permits it to have.
A federal system of government is decentralized. The United States is the supreme example of a federal system of government. Political power is split between the federal government and the fifty individual states. The governing authority of each of the fifty states is derived from the United States Constitution and each individual state constitution. Importantly, this governing authority is not dependent upon the whim of the national government. While the states may help implement and enforce federal policies and laws like the subunits of a unitary system, they also have the power to enact their own laws and policies.
A federal system poses some problems that aren't typically found in unitary systems. Citizens will be subject to a set of national laws and policies and also subject to a set of laws and policies enacted in the subunit (e.g., state or province) they are in. Moreover, crossing from one subunit into another subunit will subject the citizen to the same federal laws and policies, but a different set of local laws and policies. What may be legal in Texas may be illegal in New York. State government services and benefits in California may be different than those found in Maine.
Problems can also arise if laws at the national level conflict with laws at the local level. In the United States, the Supremacy Clause of the US Constitution generally results in federal law trumping a state law so long as the federal government had the constitutional authority to enact the law.
Federal systems arguably have some advantages over unitary systems. Decentralized power makes it harder for a government to exert undue power over its citizens. Additionally, a federal system of government allows for a level of experimentation that is harder to achieve in a unitary system. In the United States, for example, we have one federal government and fifty state governments all trying to achieve effective, fair, and just government for its citizens. Some may do it better than others and guide the rest to the best path. As Supreme Court Justice Louis Brandies has stated, a 'state may, if its citizens choose, serve as a laboratory; and try novel social and economic experiments without risk to the rest of the country.'
Let's review what we've learned. We can categorize countries into unitary and federal systems of governance. A unitary system of government is a government where political power is centralized at the national level. The political subunits, such as a state or province, are generally limited to implementing, administering, and enforcing the laws and policies that are enacted at the national level.
In a federal system, political power is decentralized and split between the national government and governmental subunits, such as states or provinces. Federal systems can be more complicated as different government bodies enact different laws and policies. Conflicts between local and national laws must be resolved. On the other hand, decentralized power helps prevent an overreaching government.
Additionally, a federal system is arguably more amenable to policy experimentation as political subunits act as 'laboratories,' experimenting with different solutions to problems without risking a disaster of national scale if the experiment ends in catastrophe. However, in federal systems, the federal government can sometimes override these experiments, thanks to things like the United States' Supremacy Clause, which generally results in federal law trumping a state law so long as the federal government had the constitutional authority to enact the law.