Schengen area

The Schengen area is the large group of European countries who have banded together to share one entry visa. When traveling as a tourist with an American passport, you don't need a visa to enter for up to 90 days (max 90 in any 180-day period in the whole region, no permission to work). Between these countries, there is normally no passport check at the border. See also http://www.ricksteves.com/travel-tips/trip-planning/travel-documents, as a starting point, and the link there to the State Department web site. Your tourist visit time is more limited than it was, say, 20 years ago, when you could go to each country for three months.

Most EU countries are in Schengen area. Your passport is checked in the first country of Schengen area you enter. Then you cross the borders of countries in Schengen area without border checks - something like between states in the USA. Details are on google under Schengen countries.

I think there are 26 European countries included in the "Schengen area."
Schengen is the name of a legal agreement or treaty that occurred (initially) in the 1980's.

Here is a map of the countries that are part of the Schengen Zone...

There is also a hefty penalty I believe if you violate the 90 day rule and if going to Italy, please see a fairly recent posting about your passport expiration date as this is very, very important.

"The Schengen area is the large group of European countries who have banded together to share one entry visa" - that is not the purpose of the Schengen agreement, it is a side-effect.
The purpose of the Schengen agreement is to abolish border controls and therefore make it easier for the citizens of Schengen countries to travel between those countries.
The side effect is that because there are no border controls, there is no system to check non-Schengen citizens when they travel between Schengen countries, so they had to implement a common scheme for regulating the entry and length of stay for non-Schengen citizens.

The Schengen agreement is sponsored by the EU and was signed in the Luxembourg village of Schengen. Most EU countries are members of Schengen (but not all, notable exception UK), plus some non-EU countries are 'in' (Switzerland, Norway etc.).
The EU has no customs between its member, so when travelling between two countries which are in both EU and Schengen, there is no border controls at all (example: France to Germany). For countries where one is in the EU and one not, there is customs but no passport checks (example: France to Switzerland). For countries both in the EU but only one in Schengen, there are passport checks but no customs (example: France to UK).

EU member states NOT part of Schengen: Bulgaria, Croatia, Cyprus, Ireland, Romania and the United Kingdom.
However, Bulgaria and Romania are currently in the process of joining the Schengen Area.
Schengen states NOT part of the EU: Iceland, Norway, Switzerland and Liechtenstein.