Schengen is a member of the Schengen Agreement.
There are no border controls between countries that have signed and implemented the treaty - the European Union (except Bulgaria, Cyprus, Ireland, Romania and the United Kingdom), Iceland, Liechtenstein, Norway and Switzerland. Likewise, a visa granted for any Schengen member is valid in all other countries that have signed and implemented the treaty. But be careful: not all EU members have signed the Schengen treaty, and not all Schengen members are part of the European Union. This means that you may have to clear customs but not immigration (travelling within Schengen but to/from a non-EU country) or you may have to clear immigration but not customs (travelling within the EU but to/from a non-Schengen country).
Airports in Europe are thus divided into "Schengen" and "non-Schengen" sections, which effectively act like "domestic" and "international" sections elsewhere. If you are flying from outside Europe into one Schengen country and continuing to another, you will clear Immigration and Customs at the first country and then continue to your destination with no further checks. Travel between a Schengen member and a non-Schengen country will result in the normal border checks. Note that regardless of whether you are travelling within the Schengen area or not, many airlines will still insist on seeing your ID card or passport.
Nationals of EU and EFTA (Iceland, Liechtenstein, Norway, Switzerland) countries only need a valid national identity card or passport for entry - in no case will they need a visa for a stay of any length.
Nationals of non-EU/EFTA countries will generally need a passport for entry to a Schengen country and most will need a visa.
Only the nationals of the following non-EEA countries do not need a visa for entry into the Schengen Area: Albania*, Andorra, Antigua and Barbuda, Argentina, Australia, Bahamas, Barbados, Bosnia and Herzegovina*, Brazil, Brunei, Canada, Chile, Columbia, Costa Rica, Dominica, El Salvador, Georgia_(country), Grenada, Guatemala, Honduras, Israel, Japan, Kiribati, Macedonia*, Malaysia, Marshall Islands, Mauritius, Mexico, Micronesia, Monaco, Montenegro*, Nauru, New Zealand, Nicaragua, Palau, Panama, Paraguay, Peru, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and the Grenadines, Samoa, San Marino, Serbia*/**, Seychelles, Singapore, South Korea, Switzerland, Taiwan*** (Republic of China), Timor-Leste, Tonga, Trinidad and Tobago, Tuvalu, Ukraine, United Arab Emirates, United States, Uruguay, Vatican City, Vanuatu, Venezuela, additionally persons holding British National (Overseas), Hong Kong SAR or Macau SAR passports.
These non-EU/EFTA visa-free visitors may not stay more than 90 days in a 180 day period in the Schengen Area as a whole and, in general, may not work during their stay (although some Schengen countries do allow certain nationalities to work - see below). The counter begins once you enter any country in the Schengen Area and is not reset by leaving a specific Schengen country for another Schengen country, or vice-versa. However, Australian and New Zealand citizens may be able to stay for more than 90 days if they only visit particular Schengen countries—see the New Zealand Government's explanation.
However, all British Overseas Territories citizens except those solely connected to the Cyprus Sovereign Base Areas are eligible for British citizenship and thereafter unlimited access to the Schengen Area.
Further note that
(*) nationals of Albania, Bosnia and Herzegovina, Macedonia, Montenegro and Serbia need a biometric passport to enjoy visa-free travel,
(**) Serbian nationals with passports issued by the Serbian Coordination Directorate (residents of Kosovo with Serbian passports) do need a visa and
(***) Taiwan nationals need their ID number to be stipulated in their passport to enjoy visa-free travel.