I've just looked at what I think you've bought, and need you to clarify that you have purchased a wireless dongle, the sort that would be plugged into a laptop to enable wireless communication with a wireless router? If this is indeed the case, then the answer is no, it will not work with any type of xbox. If your xbox is the older (phat) model, then you need to use either a standard Microsoft wireless adapter (white in colour, has 1 antenna, and will only work on 802.11 b/g routers) OR the 802.11 n model which is black in colour, has 2 antennae, and will work with 802.11 b/g/n routers - 'n' is the latest wireless protocol, and allows much greater range between a wireless router and its dependants (laptops/ phones/ xbox/ etc...)
Also, as mattclarkie said, if you have the newer xbox (slim), then it will connect directly to your wireless router.