Sure you can, does your computer have at least one extra PCI slot or a USB port? You'll find it cheaper to shop online, if you are patience you save money. Else you can always head to your local computer store and purchase it.
The PCI's are better, but requires you to do some installation, but the USB's you can just plug it in and install the driver. USB would be the easy route. Make sure you buy 802.11 g or 802.11 b/g because it's current and should be compatible with most existing wireless routers.
I know it's been a while, but will this work w/ the HP Pavilion 505N? After a little research, it seems the most common speed is up to 54 Mbps, is this correct?
It should work, but I recommend a PCI version though. Not so positive about USB, and since it's only $18 buck makes me a little nervous. Unless you have other computers you want to use it on, then it's up to you. USB would be fine.