Airspace means the air above a specific country (e.g. US airspace) but for aviation purposes it is usually divided into different types, each of which may have different regulations.
Airspace in general refers to the air - or atmosphere - above a specific country or other ground area, e.g. US airspace, Washington DC airspace, etc. For aviation purposes airspace is classified and divided in various ways; these classifications are broadly similar in most countries but the details can vary. Operating in various types of airspace usually requires the pilot to follow certain rules or have certain training, and some airspace may be completely closed to air traffic.
One important classification is the airspace class, which is lettered from A to G (class A, class B etc.) although not all classes exist in all countries. Class A is the most restrictive and class G the least restrictive. Each airspace class has different rules for who can operate in them and in which circumstances, e.g. class A in the US is limited to instrument-flight-rules flights only. See airspace-classes for questions about these classes and the differences between them.
A second classification is special use airspace, which covers many different designations including prohibited airspace, restricted airspace, military training zones and others.
Airspace classes and uses are clearly marked on aviation charts, although temporary and even permanent changes may be made at any time for security or other reasons. Understanding airspace types and their associated rules is important for all pilots.