Hello Kaine and welcome to the forums.
I think we're going to need some additional information regarding your "new computer" as well as your network setup to provide much in the way of assistance.
Please read the first post here Link Removed and provide some information regarding your new computer's specs.
Additionally since it might either be a driver issue with your wireless card or possibly a firmware issue with your wireless router we could use some detail regarding the Manufacturer Name, Model Name, Model Number and Revision Number (if applicable) for these components. And any specifics you can provide as to the current wireless driver you have installed regarding the date and version number would be helpful.
You can try some basic things like changing the channel that your wireless router is currently using as well as perhaps temporarily for testing purposes reset your current wireless security to open (none) and see if you can connect that way and then go back in to the router interface and experiment with different security settings WEP, WPA, or WPA2.
Without knowing if this is a laptop or desktop it's difficult to make suggestions. Normally with a laptop I might suggest taking it to a local hotspot to test if the problem is unique to your particular environment. A bit more difficult with a desktop.
The fact that you have another computer working fine in your present environment is of little help, unless of course the other computer is identical in every way. Same OS, same hardware and software installed, etc.,
Keep us posted
Randy