Hooking up Xbox Live

[Vampyr]

Alright, I have another question:

Here is how my DSL is set up:

A phone cable runs into a black box, which has an ethernet cable running to my computer, and that box is in turn plugged into the electricity.

Is this box a router?

If so, couldn't I just buy another one of these, put a phone jack in my room, and get another ethernet cable and then hook it all up just like my computer is?

The reason I ask is because I want to spend as little money as possible, so I just want a cheap router.