Sharing Internet Connection

I recently changed to DSL (Verizon), and it works fine. Now my roommate got a laptop computer and we'd like to share the connection (wireless, at least to the laptop). I've looked at Verizon help, Dell help, and Microsoft help and just get more and more confused. I have a Dell desktop, Dimension 2350. It has Ethernet (which I am using), but not a wireless card. The laptop is set up for wireless. We both have Windows EX Home Edition. Here are just a few of my questions.
Is there a difference between a router and a gateway? I read that, with a router, my computer would have to be on for my roomate to use the connection. But I have a neighbor who has two laptops sharing a connection with a router, and they don't both have to be on.
Microsoft says I would need two network adapters for my computer, but it sounds as if one might actually be the modem, which I already have. Some of their instructions sound to me like it's setting up the initial DSL connection, which I've already done. It says in one place that you can skip the step if you have a DSL connection, with no password. I do have a password (although don't usually use it because I don't log off.)I don't think I would need an adapter for the laptop. Right?
I have a feeling I'm in over my head on this one, and may have to pay someone to do it, but perhaps someone can shed a little light on it.