There has never been any clearly agreed and defined end point for uniting Europe. Back in the 1920's a fantastically named guy called Count Richard Nikoloaus von Coudenhove-Kalergi proposed the idea of "Pan-Europa" in the book of the same name. He, and his supporters, wanted a federal state of Europe that looked and operated in much the same way as the United States of America. They wanted a single currency, unified single government etc..

That type of thinking is called "Federalism". Probably the arch-federalist is Jean Monnet who is the father of modern European federalism. After WWII he wanted to bind the nation states of Europe together into a federal superstate by joining the economies of France and Germany. The Monnet Plan (different from the Schuman Plan) basically put the German coal producing regions under international control because the French were terrified of German rearmament.

So you have federalists like Monnet but also there were people who couldn't stand the idea of giving up national sovereignty. The typical example here is Charles de Gaulle. He saw the future of Europe as an entity where government leaders met together and decided policy but where absolutely against the idea of a common currency, joint European army and all the rest of it. Generally speaking, they wanted a European market in which the individual nations did not have protectionist trade tariffs, it was part of the strategy of speeding economic recovery and intertwining economies after WWII to make another war unthinkable and impossible.

The European project is not easily definable, it means different things to different people. An expert on the history of Europe, Desmond Dinan, has described its formation as geological. It's treaties built on treaties that didn't perhaps mean that much by themselves but as they accrued and time went on they gave rise to what we now have as the EU.

Sources:

Desmond Dinan, Europe Recast: A History of European Union, 2004.

Tom Buchanan, Europe's Troubled Peace: 1945 to the Present, 2012.