I have purchased a Router from Ebay but there is no Driver setup just a PDF file. I cannot connect because it asks for a Network Security Key, for which I am at a loss. I have tried a Network Name as I used with my recent Router and my Password for previous router, but neither are accepted.

Sounds like you are trying to setup the connection wirelessly.
Best idea is NOT to.
Plug in an ethernet cable and you should then be able to access the router.
Normally it would be on 192.168.x.1
x would be 0 or 1 depending on make and model.

Most routers come with these details on the back.

What kind of router is it? The more information the better.