The router is a D-Link EBR-2310.
It is normally connected to three computers.
It stopped working for one of the computers one day.

The computer runs Windows XP Service Pack 3.
No hardware install/software install occurred on that day.
The error is that the router is not assigning an IP address to the computer.

I tried swapping wires, and giving a working computer's wire to the router to the non-working computer doesn't change. However, giving the non-working computer's wire to a working computer causes the wire to be working.
The slot the wire was placed in didn't seem to make a difference.

Also, using a system restore back to a time when the non-working computer could access the internet had no effect.

Apologies if this is placed in the wrong forum.

That would seem to leave the NIC as the culprit.
Post Ipconfig from this machine.

It just echos "Windows IP Configuration" in response to both "ipconfig" and "ipconfig /all".

Velly interesting.
Check your network connection to see if TCP/IP is installed. Look for the adapter in Device manager.
Run a virus scan.