Suposedly, a Monetary Union (MU) is the final stage of economic integration.
In an MU, constituent states agree on a united monetary policy.
For example, imagine there are three nations: A, B, and C. A and B are in the MU, C is not. If A wishes to raise interest rates, it must do this at the same time as B. Monetary Union can be disasterous if the union is not anoptimal currency area.
The problem with lumping monetary union in as a form of economic integration is that many non-integrated economies can form monetary unions. If Japan and El Salvador are both on the gold standard, they would be a "monetary union." Likewise, before Eduardo Duhalde devalued the Argentine peso, Argentina was in a monetary union with the United States.
The most successful monetary union of all time is the United States of America. America's monetary union is overseen by the Federal Reserve.