The difference between the three main types of systems, unitary, federal, and a confederation, is the amount of power each system gives to the central government.
In a unitary system, the central government is all-powerful, and has the last say in matters. A unitary system can still have regional and local governments, like Great Britain and France, but the central government oversees these regional and local governments, and their powers are given to them by the central government.
A federal system is a system in which the central government and regional and local governments share power. This is the type of government the United States has. The Constitution of the U.S. gives certain powers to the central (federal government), and gives other powers specifically to state governments. The states have a lot of power to govern themselves, particularly when it comes to laws regarding individual citizens. However, in other areas, the federal government holds the power (e.g., the power to make treaties and handle diplomatic relationships with other countries), and the states have no power in these areas. Other countries with federal systems are Brazil, Germany, Australia, and Malaysia.
A confederation is a government where the states or regional governments have more power than the central government. Each state/regional government has all the powers of an individual country--they can raise a military force, print their own money, handle diplomatic relationships with other countries, tax their citizens, etc. The United States was a confederation under the Articles of Confederation. This governmental structure failed because the central government was too weak to effectively function, and the Constitution made the United States a federation. The former USSR, now known as the Commonwealth of Independent States, is structured as a confederation.