The above is correct.
Short answer: ethernet connections are the network standard. Have been for years and years. For some stupid reason (cheaper?), the modems supplied by ISPs like Pipex tend to use USB connections. USB isn't compatible with ethernet (I think some have found adapters of a sort, but they won't work properly).
So, you need to get yourself a router and modem in one unit.
http://www.ebuyer.com/product/52244
That's an example product that will do what you want. Modem and router combined.
Connect router to your phone line, and then each computer to the router (it has 4 connections on it).
It's wireless too, and you can get just wired ones (no wireless capability), but the wireless are as cheap now, and one day you may wish you'd gone wireless.
The router can be easily set up by going to 192.168.2.1 and entering your default router password (admin:password probably). This is all in the manual though.
You can turn the wireless off, if you don't want to use it. Then you'll need no password (just change the router's password itself, for the admin pages, and write it down. But you won't need that each time you connect a computer).
It should then be as easy as connecting the router, leaving for a minute until it establishes a connection, then connect or turn on each computer. Both should see the internet easily.