while technically correct, wireless network adapters are just called wifi or wireless adapters, and network adapter refers to the wired / cabled kind.
without knowing the details on your desktop, all I can say is purchase the least expensive name-brand adapter you can find. Linksys, Belkin, Netgear, etc - they're all good and they're all built by a Chinese company called Realtek
you don't need a dual band adapter, those are used for very fast connections like fiber optic or cable modem where one wireless band can't transport the full speed of the connection. or if you live in a very busy apartment building and all the lower wireless freq are full.