Schengen Area
The Schengen Area is a collection of 29 European countries (many of them members of the European Union) who have no border controls between them. There are, however, exceptions included below.
The Schengen area has fully rolled out the EES System to track all border crossings. You can check your ETIAS status here. More on crossing borders in the Schengen area here.
US citizens are allowed to stay for up to 90 days within a 180-day period.[1]
Those who have residence, work, or study permits in Schengen countries are often given Schengen visas affixed to their passports allowing multiple entries into the area for a year to allow time to get a resident card.
Exceptions to Schengen
Some countries have put "temporary" pauses on their Schengen agreements which allows them to conduct passport checks at borders and within their borders at random. The reality is that these temporary measures are more or less permanent now, are usually based on exaggerated security fears, and are now being used as a political cudgel between sparring states. You can see the current list of exceptions here.
European areas not in Schengen
Some EU countries are not in the Schengen area so they will require passport checks. These include Ireland and Cyprus. Some areas within the EU like the Spanish cities of Ceuta and Melilla are also outside the Schengen area.