Many large enterprises use computer technology from different companies (IBM, Microsoft, Oracle, Sun etc), and it is often difficult to get these systems to connect and interface to each other.
Middleware is the connecting software that help these systems to work together.
One example is messaging.
If computer A wants to send a message or information to computer B it is not always that easy if they run different technologies.
Computer A wants to be certain that computer B gets the message before computer A deletes it.
So you need messaging software that run on many different computer platforms that sends and monitors these messages.
So the messaging software will send the message from Computer A to Computer B, but then wait until Computer B verifies it has got the message before deleting it.
Middleware is often the unsung hero of computing.
When you draw money out of your banks hole in the wall machine it is middleware that sends information from the computer in the local bank to a central computer which debits your account by say �20 before giving you the money.
If for some reason the local banks machine cannot deliver the money to you it is VERY IMPORTANT the central computer does not debit the �20.
This is where secure messaging middleware comes in.
More here
http://en.wikipedia.org/wiki/Middleware
http://www.sei.cmu.edu/str/descriptions/middle ware.html
http://www.middleware.org/