A federation is a union of a number of self-governing states or regions, which are joined together under a central government. The central government is not able to do a lot of things that central government in more centralized states can. These things are done by the states (or regions) in a federalized state.
The United States of America is a federation. The federal (central) government has the highest executive role. However, state governments can exercise any powers as long as those powers are not assigned to the federal government (by the United States Constitution) and if those powers are not prohibited by the states themselves.
Australia became a Federation in 1901.
After the 1971 civil war, Pakistan became a Federation adopted in the 1973 Constitution of Pakistan.
Other examples of federal states are Austria, Belgium (since 1993), Canada, Germany, Russia and Switzerland.