The model of the motherboard. Try copying down, or photographing, every sticker.
But if you have access to another machine, or a friend with another machine, you could download one of many utilities that will tell you about the mobo, put it on a flash drive, and run it on your machine.
Or, perhaps, the Intel Driver Update utility can be downloaded in its entirety? I doubt it.
Go to Device Manager ( right click on My Computer, select Properties, go to the Hardware tab, launch Device Mangler), open up the section for Network Adapters, look for your network devices, and send us the descriptions. You can right-click on one to get more information.
Is there a yellow exclamation point next to the network device in the Device Manager? I'm wondering if the problem is actually the driver. The port could be dead, the cable could be bad, and so forth.
If you have a CD that came with the motherboard, it may have the network drivers on it. Do you have the CD?
Did you get this machine second-hand? Is it homebuilt or a manufactured system? If it's a manufactured system, say a Xell 233-R, you would go to the manufacturer's website on a friend's machine and download the drivers.