I supposed you'd call it a live attendant. I'm sure you're all familiar with being on a site and being able to click a button to initiate a chat.
I've also seen some time ago, though, a service where you don't need to wait for someone to initiate chat. You can see visitors pop up in a panel and start chatting with them and asking them specific questions about what they're viewing.
I've done google searches for live attendants and chat features for web sites but it seems like all services I find are the former and not the latter.