A federation is a state composed of states — a union that is neither a single nation nor a collection of independent countries. It is a compromise, and like most compromises, it is unstable.
Federations dissolve when the central government can no longer hold the constituent units together. This can happen suddenly (the Soviet Union, 1991) or slowly (the Holy Roman Empire, which took several centuries to fully dissolve and was, by the end, neither holy nor Roman nor an empire). The proximate cause is usually economic: the centre can no longer pay for the loyalty of the periphery. But the deeper cause is always the same: the benefits of union no longer outweigh the costs of compromise.
The dissolution of a federation is rarely clean. Borders must be redrawn. Debts must be apportioned. Armies must be divided. Citizens who woke up in one country go to bed in another. Some federations dissolve into their constituent parts (Czechoslovakia, 1993). Others dissolve into chaos (Yugoslavia, 1991–2006). The difference is largely a matter of whether the dissolution was negotiated or imposed.
See also: the dissolution of the Soviet Union (1991).