Where does it lose you?
You know what an operating system is?
You know what client-server architecture is?
My read is that middleware sits between two components that are usually thought of as talking directly to each other.
Actually I think I may be able to describe the real-time compositing software I have as a kind of middleware that does not involve distributed computing.
Basically - the OS uses a driver to make my camera available to programs like skype.
I wanted to be able to edit the image on the fly so I could appear to be calling from space, the top of a mountain, in Max Headroom's TV, etc.
So this software stands between the camera driver and skype basically by telling the OS that it is a second - fake - camera. I tell the skype to use the fake camera and the program to use the real one - now it plays man-in-the-middle.
You can do similar stuff with server-client interactions ... like a bit of malware or AV-filter that sits between your email client and your wifi card. Normally the email client talks direct to the email server ... but now the email client talks to the filter (or whatever) and the filter talks to the server.
wikipedia has other examples.
think of it as a buzz word.