I use a western digital my book world. You plug it into your wireless router by an Ethernet cable. You can then access it from any computer on the network, wired or wireless. Or even over the Internet if you set it up properly. It plugs into the mains, so doesn't require a computer to be on.
Transfer speeds aren't enormous- I think it originally took about 40mins+ to move over 600gb by wire, and it would be slower by wireless. But it lets me stream video and music (including 1080p blue ray rips) without any problems to my pc and ps3 over a wire, and to my iPad (though that plays 1080p at 50% speed, but I suspect it's the fault of it's puny little processor not the wireless connection).
If I understand your question correctly, yes you can access the NAS wirelessly and move around data/ stream content to your pc whilst also using the Internet etc. Though they will obviously compete for bandwidth.