Googling this question returns lots of results with marketing jargon.
What I am looking for is a summary of what it does, more along these lines What is Maven?
"WebSphere" in the most abstract sense is a brand encompassing various products, so "a brand" is about as good an answer as you can get.
What one usually refers to when saying "WebSphere", though, is the WebSphere Application Server, which (as the name implies) is an application server - something that runs EJBs and other server-side Java technologies like servlets, JMS, etc. An example of another important player in this space is JBoss AS.
Finally, another possible definition of "WebSphere" is "its creator's biggest technology mistake" ;)