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You need a modem whether you use a router or not. If decide to use a router you'll need an external ethernet modem that works with your ISP.
The function of a router is so you can connect multiple computers to one internet connection and setup a lan if you choose to. If you only have one computer you don't really need a router unless you want the NAT protection that it provides(good thing).
You will still need a nic (network card) in each computer.
So in essence you will have your phone line (DSL) coming to the modem, a cat5 cable to the router, and a cat5 cable to each nic. If you don't use a router the cat5 cable from the modem will go to the nic in your computer.
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