How do the countries work together?

The EU is based on treaties between its member countries. These set out what powers the EU has, and how it can use them.
These treaties have to be agreed by the governments of each member country, and by either its national parliament or its people (in a referendum) or both. Once member countries have agreed these rules, they are all expected to stick to them.
The first treaty was the Treaty of Rome, which created the European Economic Community in 1958. Important treaties since then include the Single European Act (signed 1986) which set out to complete the single market, and the Treaty of Maastricht (signed 1992) which turned the Community into the European Union and created a common foreign policy.
The Treaty establishing a Constitution for Europe (“the EU Constitution”) is the latest in this series of treaties.

http://europa.eu.int/abc/obj/treaties/en/entoc.htm