Java EE is a collection of specifications for developing and deploying enterprise applications.
In general, enterprise applications refer to software hosted on servers that provide the applications that support the enterprise.
The specifications (defined by Sun) describe services, application programming interfaces (APIs), and protocols.
The Java EE product provider is typically an application-server, web-server, or database-system vendor who provides classes that implement the interfaces defined in the specifications. These vendors compete on implementations of the Java EE specifications.
When a company requires Java EE experience what are they really asking for is experience using the technologies that make up Java EE. Frequently, a company will only be using a subset of the Java EE technologies.
A reference implementation of Java EE is GlassFish