The EU is a political and economic union of 28 countries, which developed an internal single market through a standardised system of laws. Politically, these 28 countries are linked by the European Parliament, the European Commission, the Council of Europe and the European Council (the last two have similar names, but they have different functionalities). Economically, by the ECB - the European Central Bank. Monetarily, by the Eurozone and the currency EURO, which is present in 19/28 countries. Also, there's a Schengen area, where passports are abolished and you can travel freely, BUT not all the countries that are in the EU, are in Schengen area (ex. Romania, Bulgaria); and vice-versa: Norway is not in the EU, but is part of the Schengen area.
I hope I helped, I used to have a "map" of interactions of all these countries in different unions/institutions, but cannot find it. If you need one to help you understand the EU better, tell me and I'll intensively look for it!
Update: here's the map.