I recently acquired a gift subscription of National Geographic for a close friend and received a lovely map of the world. When I looked at Europe I saw that countries like France, Germany and Belgium were still shown as being separate rather than joining all their outlines and labelling it as “The European Union”. Why? I thought that those countries are just states or provinces or prefectures or whatever the diminutive of “country” may be in this context.
The European Union has some administrative powers, and many (but not all) Union members share the euro currency, but member nations are still sovereign.
The European Union is an economic entity, not a geographic one. The member nations all retain their own laws, parliaments, etc.
You might as well say that the map still shows the U.S. and Canada and Mexico as separate countries, even though they all belong to the United Nations. Similar idea.
Erm, ditto. Think NAFTA, or the now-ghostly CIS (which did appear on some maps I’ve seen as one blob), and it is closer to the spot. There are actually a number of European organizations, including a defensive military alliance similar to NATO, but restricted to Europe (the name slips my mind right now, bleh).
On the other hand, many people do consider the EU and these other organizations steps towards making Europe a single entity with member-nations… Similar to how the United States started (with each state regarding itself as independent and entering into economic and military alliances with the other states). I’ve seen this used as both for and against it. 
European nations, unlike the American states, however, all have very long histories, prejudices, culture, concepts of individuality, political, and economic viewpoints, so don’t expect to see Europe as a single blob on a map any time soon. Frankly, if you can see France, Germany, Italy, Poland, etc smiling happily under a single flag in front of their united government building, you need to lay off the drugs. 
The European Union has a larger scope of action than the UN or the NAFTA .
It´s more than a free trade association - though it started out as one, way back when… first came the European Coal and Steel Community, establishing a common market (w/o tariffs) for coal and steel between 6 countries, later the European Economic Community (EEC) etc. - but it was always more than that, there was a common defense agreement which later developed into the Western European Union (WEU), which still exists today (Zagadka, I think you mean that one) (read up on the full version at http://europa.eu.int/abc/history/index_en.htm)
To make a long story short, the EU member states today cede part but not all of their sovereignity to the Union, especially in economic matters (common market, monetary union (though not all EU member states have the EURO), import/export regulations), but also in legislation - environmental protection, industrial regulations, pharmaceuticals, transport etc. However, the member states are allowed to have tighter regulations, the Directives and Regulations are a sort of minimum standard. eg. emission control - there are Directives that fix limit values for pollution, but some countries have even tighter controls. Or foodstuffs must be labelled in a different way, because some countries have stricter regulations.
There is a lot of exchange at University level, but every country has its own educational system. The member states retain sovereignity in most areas of domestic legislation and policy (eg. police force).
The difference between the EU and the UN is, that in the UN, the states don´t cede their sovereignity in any way. The UN has no legislative power.
But as the others said, the EU is far from being a superstate or federation (although some do strive for one - won´t become one for quite a while, though).