your new computer, we have no way to know it it has any sort of appropriate expansion slot. "Hosting a graphics card" is not required.
If there is an available PCI slot - or in the case of a laptop, an available PCMCIA or ExpressCard slot - then you can move forward with a firewire card.
If there is no expansion slot, another option in addition to Bob's suggestion, you could also do an external analog/digital converter. This method connects the camcorder's AV-out to the analog/digital converter and the A/D converter plugs into the computer's USB port. Elgato, Roxio and many other make these. The step down to analog and compression through USB will reduce quality a little - but without the computer firewire connection, choices are limited.