ADSL Modem:
This is very similar to your standard dial-up modem, it's a piece of hardware which connects to a single machine which then allows that machine access to the internet, any form of internet connection sharing must be done on the machine accessing the internet.
ADSL Router:
This is effectively a network switch with a built-in ADSL modem, the switch does all the dialling in and sharing of the internet connection to those machines connected to it, ADSL Routers also tend to come with a firewall (stateful packet inspection or NAT) and allow you to configure them via either Telnet or a web interface. There's no need to set up anything on any of the PC's on the network (other than telling them to look for an internet gateway on the network), the Router handles it all.
Over POTS:
POTS - Plain Old Telephone System - this is what you, me, Telkom, whoever uses as a telephone network, it consists of copper cabling and all the other funny telephone system stuff that Telkom doesn't want you to know about - This just means that your ADSL connection functions over a standard telephone line, much like most ADSL installations Telkom does.
Over ISDN - This I may be wrong with, correct me if it is, I'm taking a stab in the dark.
This, similar to Over POTS, utilizes any ISDN technology you may have - this is most useful for those who have ISDN and are now looking into "upgrading" to ADSL instead of having to have everything removed and replaced (thereby being costly to the end user), other than that, there's little in the way of added benefit.