Real Quickly: What's the Web?
Let's start with a cliche' one line description of what the World Wide Web is. It's basically a lot of different files (all over the world) that are linked to each other, so that you can look at a file that has a link to another file and then follow that link to read the next file. (These days, it's not just files but useful programs, too.)

What makes this so powerful is that these files can contain graphics, or snippets of animation or music, and they can contain information that you normally would have to find with special programs like ftp or gopher (which is fine for nerds, but is a real pain otherwise). Or they can let you use statistical programs, user surveys, games, and so on. These days all this is taken for granted, but pre-1994, things were rather different!

There. Now on to things that are more interesting, like ...

What's the deal with Internet Explorer, Netscape, and Web Servers?
OK, so I have to use a little jargon.

Internet Explorer and Mozilla allow you to view World Wide Web files. They provide the pretty little window with all the neat buttons. Other competitors that do the same thing (basically) are things like Netscape, Mosaic, Opera or LYNX (LYNX is pure text, and hence not much fun unless you're stuck dialing in over a phone line and hence can't get graphics anyway). Explorer and Netscape and their kin are clients, of a type called web browsers (just a type of computer program). They're like different types of TV sets that show you what's being broadcast. (Not a great analogy, but there you have it). To exten