This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
Web servers, like Apache, typically just map request to files on the filesystem. So, serving up an html file, image, mp3, or other things that can simply be requested, fall into the domain of a web server.
WebSphere is an application server that implements logic through Servlets and JSPs. An application server can 'think' about a request, and deliver customized content to the end user. Web servers aren't smart enough to do this.
The web container of the WebSphere Application Server runs on an http port, so it can do the job of a web server, but typically, an apache or IHS web server sits in front of the WebSphere Application Server.
Here's a bit more info on the realtion between WebSphere and a web server: