The motherboard: Motherboard Specifications, N-Alvorix-RS880-uATX (Alvorix) HP Pavilion p6623w Desktop PC - HP Customer Care (United States - English)
is spec'd as having a PCI-E X16 graphics slot, "Gen 2 speed". That should be compatible with any PCI-E X16 graphics cards, including the new PCI-E 3 cards.
The PC is listed as having a 250W power supply, so that'd need an upgrade for anything but very low-end graphics cards.
I disagree about getting an ATI/AMD card to match the AMD CPU. Get the make that you'd like. As far as I know, there are no compatibility issues. Some of the newest AMD motherboards have been licensed by nVidia to support SLI.