ESTA card requirements when traveling in Europe

ESTA authorization is a requirement for certain visitors to the US. As a US citizen, you are not able to get it and you do not need it.

Furthermore, because ESTA concerns visitors to the US, it is not helpful for people traveling in Europe.

The ESTA is a US system where many visitors coming to the US are required to get an electronic authorisation in advance. The US insists that this authorisation is not a visa and it does involve much less bureaucracy than a traditional visa but nevertheless it allows the US authorities to check out travellers in advance and reject ones they don't like. A US citizen cannot get an ESTA and there is no such thing as an "ESTA card".

The Schengen area is planning to introduce a similar system called ETIAS but the system is not yet active. Current plans appear to be for this system to start in 2020.

So for the moment as long as they follow the 90/180 rule US citizens can make tourism and business visits to the Schengen area without having to get any form of advance authorisation from the European authorities. They will obviously need to carry their US passport. The non-schengen EU countries also currently allow US citizens to visit without advance authorisation.

P.S. Canada and Australia have introduced similar systems but US citizens are exempt from the Canadian one.

There's no such thing as an ESTA card in Europe. You will be given a 3 month stamp in your passport as a US citizen when you enter your first Schengen country from that list. There will be no other passport controls on your trip as all these countries are in the Schengen zone.