You probably should call your DSL company and ask them what to do.
Usually you cannot just switch the cable from one computer to the other because each computer has a unique IP address and the DSL company will see that a different IP address is being used and will not recognize it.
The way around this is to have a router hooked up to the modem. The router has its own IP address and that is what the DSL company sees. The router has several ports on it for hooking in computers using ethernet. A wireless router does the same thing except that it also allows computers to hook up to it using wireless.
A hub is like a router except that the hub does not have its own IP address and so the DSL company sees the IP addresses of each computer hooked up to the hub. Unless required by the DSL company, a hub should not be hooked up directly to the modem (a hub can be hooked up to a router to allow more computers to be hooked up).