Will my computer support an ATI Radeon series graphics card?

Yes, check if you have PCI,AGP or PCI-e. Also, be aware of PSU requirements, the space inside your tower (you may have to remove adjacent cards like a modem/sound card) and if you have sufficient cooling...Almost all modern graphics cards are PCI-e and if you want to play todays games you need to have a PCI-e slot. If not, you could buy a new motherboard (I suggest an SLI ready) but that is not really worth the hle if you don't know what you are doing or don't want the hle. Instead invest in a new PC. And a side note, some games require a dual core processor (asins creed comes to mind) and all games will take up memory-So you will need to have sufficient RAM to prevent your computer from lagging outside of the game..