There are no border controls between countries that have signed and implemented this treaty - the European Union (except Bulgaria, Croatia, 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 there may be spot customs checks but no immigration checks (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, but not Customs, at the first country and then continue to your destination where your baggage will have customs checks but there will be no further immigration controls. Travel between a Schengen member and a non-Schengen country will result in the normal border checks. Regardless of whether you are travelling within the Schengen area or not, many airlines will still insist on seeing your ID card or passport.