Well, quite simply, .exe is the extension for an EXEcutable, or Application, on a Windows computer (the extension under OS X for an application is .app, BTW).
An .exe cannot be run on a Mac because it is compiled for an X86 processor, the processor a PC uses, not the PowerPC processor that a Macintosh is built around.
You are able to open a word (.doc) file (and many others: .txt, .html, .rtf, .gif, .jpg, .bmp, for example) on both Mac and PC because they are not compiled. In other words, they are not Applications. They are merely data that an application, running on either Mac OR PC, can interpret.
Not to complicate things, but there are some applications that can be opened both Mac and PC (and others: Linux, etc.). These include applications coded in Java. The reason for this is that they are also interpreted, not compiled (part of the reason Java apps are so slow). To run a Java program you must have the correct Libraries installed for your OS.
a_iver: That is not quite correct. Darwine will not run .exes without emulation. In order to run a .exe you must have an X86 processor, so Darwine will have an X86 emulator built in. I'm not sure how they'll do this, but if you go look, it's on their site.