Place the PC with it's right-hand panel down on a sturdy work surface. This is because you will need to remove the left-hand panel to gain access to the PC innards. Judge left and right by looking from the front of the case, and using an appropriate screwdriver, remove the retaining screws from the back of the case, laying the panel to one side. Now, before you actually touch the card inside, ground yourself in order to get rid of any static electricity you might have stored up – static electricity is very bad news for delicate electrical components! To ground yourself, touch a metal facing of your PC or a nearby radiator. Alternatively, wear an anti-static wristband which will draw electricity away from you and funnel it through the PC case.
Now, locate the graphics card and remove the retaining screw from the graphics card backplate. The graphics card is typically the top-most card in the PC. To make doubly sure, check the back of the PC and look for the D-SUB connector – this is the blue socket where your monitor plugs into. If the two are level, you're in the right place! Grasp the card by the topmost edges and pull the card out of the slot – try to pull the card out in a straight 90-degree angle to avoid damaging the slot.
Take the new card out of it's anti-static packaging and again holding the card by the edges, line up the card with the slot and push down firmly, making sure the card goes all the way into the slot. Don't be scared to use a little force if required as the fit might be quite tight. Once you have the card firmly in the slot, screw it into the backplate and attach any power fittings the card might require. Be firm, but gentle when attaching the power fittings as even the slightest knock can cause the graphics card to loosen slightly or not fully connect with the motherboard, causing your PC not to boot. If this happens, loosen the backplate slightly and realign the card, again making sure it's fully in the slot before tightening the screw again.
Now that the card is in place, reconnect the PC and switch on! It's often best to leave the case-side off while you finish the installation, just in case you need to troubleshoot the installation if it doesn't work. Once the PC has powered up and logged into Windows, the PC should automatically detect the new hardware and begin the installation wizard. If you downloaded the latest driver for your card from the Vendor site (either ATI or Nvidia) then you'll be able to skip the wizard, and browse to the location where you downloaded the drivers. Double-click on the driver file and the drivers will self install. Reboot the PC when done, and all being well, you're new card should successfully complete the installation automatically.
All that's left to do is reattach the side of your case, and enjoy the new graphics muscle your PC now possesses! Although, remember, if your graphics card supports it, it might also be beneficial to download the latest version of DirectX so that your new graphics card can take full advantage of the new features.
Congratulations! You're done!