There are three main contenders you should be looking at Sony PlayStation 3, Microsoft Xbox 360 and Nintendo Wii.
Nintendo Wii and Sony PlayStation 3 offers wireless as an option out of the box, if you want to make the Microsoft Xbox 360 wireless you have to buy an extra component. If you don't want to use wireless or don't have wireless at home then Ethernet is the way to go. You set the console up pretty much like any network device, you can use DHCP or manually enter IP, DNS etc...
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