Websphere, the middleware product from IBM is the majority system I have been working with. After a few years commercial launch. My team tends to replace with freeware system which is JBoss middleware in order to save the license cost.
In my mind, Middleware was the most complicated thing to understand due to the limitation knowledge of people. Only large IT company will use the middleware for their applications. They use them to manage and dedicate resources for various online applications. It also uses for doing redundancy to be able to online failover and decrease the risk of system failure.
Database access services are often characterised as middleware. Some of them are language specific implementations and support heterogeneous features and other related communication features. Examples of database-oriented middleware include ODBC, JDBC and transaction processing monitors.