Well, it works well with Apple's AirPort basestations, but the PowerBook wouldn't be that useful if you couldn't take it the local café and connect to their "generic" network... 😉
You do need a wireless router. But it can be any standard wireless router, even a really old one. A "modem" is something that is used to connect your computer (or network) to the Internet, through your service provider. In the old days, they connected through the phone line and made annoying sounds. Most users have a "broadband" connection these days, and you often "rent" the modem from your service provider. Some more recent ones have a built in wireless router. Do you have a more recent Mac or Windows PC that connects wirelessly? How does it connect to the Internet?
When you click on the AirPort icon in the menu bar, do you see a list of nearby wireless networks? (First, make sure AirPort is turned ON.) Do you see the one you use with your newer computer? If you do, what happens when you select it?
could I use a USB wireless adapter instead?
You could, but you would still need to connect to your wireless network with it, which is basically the same way you connect with your built-in AirPort card. It's actually more complicated with a USB wireless adapter, because you have to install drivers and software to make it work, and that software is often not compatible with your old version of Mac OS X.