Should I use a modem or router?

This depends upon how you utilise your broadband connection. For basic browsing and emailing, a modem will perform perfectly adequately as the processing requirement is bursty.

Applications that require constant activity will benefit from the use of a router, as relocating the processing task to a dedicated router allows the PC to get on with the important job of processing information. 

The primary advantage of an ADSL router is for situations where you need to have more than one computer connecting through the same ADSL connection.  You can then connect all your computers to a LAN (Local Area Network) switch, and then connect the switch to the ADSL router - giving Internet connectivity to all the switch connected computers.  Each computer needs an Ethernet card installed and a straight though UTP cable connecting to the LAN switch.  Alternatively, there are 'wireless' networking alternatives available, either utilising a wireless ADSL router or wireless hub with each computer having a wireless network card.

If you were attempting to use an ADSL modem to allow other computers to connect to the Internet, you would have to use some form of software based 'connection sharing' on the computer that had the ADSL modem attached to it, and other computers would connect to this one for their Internet connection.