An application server is a server that is designed for or dedicated to running specific applications. At its most basic, it might be used to run one application. If that application is the one that keeps a company network going and is, therefore, a massive application, it might take up the entire RAM and ROM requirement of one server.
Another possibility is that a server is used to run certain kinds of applications. For example, a company might have several word processing, spreadsheet, or desktop publishing programs, and all of those applications might reside on one type of server. Everyone who needs to access those programs would then log on to the Desktop Publishing Server, for example, to use whatever design program that the company might recommend and have on hand.
Another kind of application server is one that runs an operating system. This is more old-fashioned, but it is still used. Certain computers, more commonly called terminals, connect to a server in order to access basic functions.
Like ants and bees that have specific roles in life, application servers have specific parts to play in the overall anthill or beehive of a company. As with many things computer-related, there is a more complicated example. Oracle, one of the main server developers, has come out with a J2EE (Java Platform Enterprise Edition) server. A huge number of companies worldwide use the J2EE server, which is able to run very powerful applications that can be accessed by multiple computers connected to that server on a network.
What it all boils down to, though, is the computer world's propensity to designate certain resources to perform certain actions. In the same way that someone might use a word processing program to write a paper but a photo manipulation program to touch up vacation photos, tech departments use application servers to compartmentalize the way that a company's multiple computers do certain things. If they're doing their job well, most people will hardly be aware that such activity is going on.