The card would almost certainly have been a network adapter (card) and as your new computer already has one, you can connect with that. You can tell quite easily because, if you unplug the cable from the old PC, its "RJ45" plug should fit into the new PC's network port.
Much less likely is that the old PC didn't have a USB port and they installed a PCI card with one or more USB ports, so that the Cable Modem could connect via USB.
NOTE: To swap over from the old PC to the new one, you should power off the Cable Modem and leave it without power for a couple of minutes. This is important so that it "releases" the association between the existing connection and the IP address it allocated to the connection. It will then detect a new connection and allocate a new IP address to it. Power on the Cable Modem and wait until it has booted fully with all usual lights displayed, then boot the PC.