To use computers, users need certain software to be present on those computers. These software are known as applications. Common examples of Applications are Microsoft Word, Lotus 1-2-3, CodeWarrior, etc.
A server is a huge, powerful computer into which users can login and access the resources provided by the server.
Thus, an application server is a computer which allows users to login and work with certain applications on the server itself, without having to install those applications on the user's home machine.