https://en.wikipedia.org/wiki/Template:Supranational_Europea...
https://en.wikipedia.org/wiki/Schengen_Area
https://en.wikipedia.org/wiki/European_Union
https://en.wikipedia.org/wiki/Eurozone
Ireland and Cyprus are in EU & Eurozone but not Schengen; Poland, Hungary (and more) are in EU & Schengen but not Eurozone; Switzerland is in Schengen but neither EU nor Eurozone; Montenegro and Kosovo are in the Eurozone but neither the EU nor Schengen.
You are required to have a passport (or id) with you (as in, that’s what the law says). Even in your own country. But in your own country a drivers license is usually also sufficient.
But in practice you will almost never be asked to show any of those. In your own country, nor abroad.