There are different laws in different European countries, so you would need to look at some individual countries rather than the whole continent. There are also differences between countries inside and outside the European Union, though countries within the EU don't all have to follow the same rules either.

You could start by getting some general information for Americans travelling abroad. After that you need to look at specific countries - for example, if you put 'american citizen work in italy' in a search engine, you will get to the information you need. But it's safe to say that an American in any European country will need a work permit and a specific visa, and that most countries have tightened up their entry requirements.

It is still, on balance, generally easier for Americans to live in Europe than the other way round if you don't have a job already lined up or are able to put money into the country.