piersk - you are indeed confused 🙂
ADSL is DSL. DSL is the generic term for all "Digital Subscriber Lines". They include RADSL, IDSL, ADSL VDSL etc. They all work in the same way, but the exchanges are set up differently. These different standards exist due to the differences in phone lines/distances DSL is expected to work over. RADSL, for example, works better for long distances (but is slower).
No matter what sort of internet connection you have, be it dial-up, cable, DSL (of any sort) - you can put a router between the outside world and your network. If it's dial-up you'll need a special router/modem, but it is easily done.