You only need that when your Xbox isn't Wireless built in or you don't have router.
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you need a special adaptor
You don't. If you have the new Xbox 360, you don't need a wireless adapter because it has built in wifi. But if you have any other Xbox 360 console then you will need a wireless adapter.
No, you cant get on the internet if you dont have it.
No it doesn't. To enable wireless network connection you will need to buy the same adaptor that is used on both the 60GB & Arcade consoles.
You either have to have the wireless adapter or the xbox 360 slim (which is better)