You can move freely amongst counties in the Schengen Area, which is a subset of EU counties (which itself is a subset of European counties).

> The free movement of persons is a fundamental right guaranteed by the EU to its citizens. It entitles every EU citizen to travel, work and live in any EU country without special formalities.

Source: https://ec.europa.eu/home-affairs/what-we-do/policies/border...

As an aside, this was a hotly contested subject during the UK EU “Brexit” referendum.