I still meet a lot of people who are confused by the term ‘WebSphere’. Let’s clarify, as it has two oft-used meanings. The first, which IBM encourages, refers to the WebSphere brand of software. This comprises many products (probably over 100 once all variations are taken into account), most of which are somehow related to middleware or integration. Included in this brand is the product WebSphere Application Server (WAS), upon which several of the other products are based. The second meaning of the word is as a synonym for WAS, and is popularly used outside IBM (and sometimes inside, when IBMers are feeling lazy). If someone asks ‘do you know WebSphere?’, they are almost certainly using the word to refer to WAS specifically.
For more information, see the Wikipedia entry for WebSphere.