What is Application Server?

Legal Definition
An application server is a software framework that provides both facilities to create web applications and a server environment to run them.

Application Server Frameworks contain a comprehensive service layer model. An application server acts as a set of components accessible to the software developer through an API defined by the platform itself. For Web applications, these components are usually performed in the same running environment as their web server(s), and their main job is to support the construction of dynamic pages. However, many application servers target much more than just Web page generation: they implement services like clustering, fail-over, and load-balancing, so developers can focus on implementing the business logic.

In the case of Java application servers, the server behaves like an extended virtual machine for running applications, transparently handling connections to the database on one side, and, often, connections to the Web client on the other.

Other uses of the term may refer to the services that a server makes available or the computer hardware on which the services run.
-- Wikipedia
Legal Definition
A computer that has app programs to serve the client computer when the request is put in by other network users.