Two choices:
1.) Find out what kind of video card you have right now (most likely "integrated" with the motherboard, which are alright for most basic functions but many games need an independent one) by using the System Information feature in Windows XP (Start > Programs > Accessories > System Tools > System Information). Under the "Components" section, look for "Display" and the first item on that list should be your graphics card. This is the painless part.
Next, you'll need to find out what kind of graphic-specific inputs you have available on your motherboard (since you've bought it around a year ago, I suspect that it will most likely have at least a PCI and/or an AGP port, possibly even a PCI-E) . You might consult your PC's manual and see if they have a diagram of the motherboard (many manuals do). You could also contact the manufacturer of the PC who should be able to identify your model (especially a Dell, since it has the service/ID tags right on the case). Since you're feeling computer illiterate, this last option may be kind of out of the question, but you could carefully open up the PC and check the insides yourself, and determine what you've got. The problem is you might not know what you're looking for.
Ignore the other cache thing. Those long white inputs are called PCI (Peripheral Component Interconnect), which is where you would insert a graphics card, modem, etc. into your computer so that it will interact and be functional. You most likely have a couple of these already occupied in your PC.
However, after awhile, it became necessary to speed up the interaction between the motherboard and the graphics card, which is why...
The Accelerated Graphics Port (or AGP) was created, so that it could communicate exclusively with a graphics card and process information faster than a generic PCI port. The AGP port is exclusively used for graphics cards, which makes this process able to work (PCI, as aforementioned, can be used for modems, sound cards, etc.). The AGP port is in maroon. Note that the "teeth" or input of the AGP is different than the PCI, which means that you have to be careful when buying a video card, since they are specific to the type of input (i.e. if you buy a PCI video card, it will not fit in an AGP port and vice versa). Don't worry though, this is usually designated right on the front of the box, and many models are available in either version.
Which brings us to our last port:
PCI-E (or Express) is an update on our previous PCI component. Again, look at the relatively unique input structures. Its even faster than AGP, and is not exclusive to graphics cards. This is usually not a feature you'd ordinarily find on a computer out of a box, and you probably don't have it. Its relatively new, and is pretty much reserved for hardcore programmers and gamers at this point.
Anyways, once you determine the type(s) of ports that you have, select the *best* one from the list I mentioned (probably an AGP in your case). Hit the local shops, and see what they recommend. Prices will vary, and don't necessarily reflect the quality or performance capabilities of the product (
this is a good example - one site lists it for about $250, the other $95 :p ). I'd look to see what EA recommends, and go from there.
Once you've purchased the card, you could either hire someone (like a Geek Squad from Best Buy or Circuit City) to install it for you, or you could do it yourself (which is fairly simple, I could walk you through it with the online chat).
2.) Buy an X-Box 360 or PS2 and just return your PC version of the game. The graphics card you might have to buy would probably be pricier than buying a PS2 right now, so it might be a *better* option to just get the PS2 (which is also a fine DVD player, CD player, and has a huge game library) and the PS2 version of the Godfather game rather than trying to upgrade your PC, which could become costly, for one game.
Or, you could shell out considerably more cash and get the X-Box 360 and that version of the Godfather, which has superior graphics, but costs more.
lol
loyalty and pay my MAXIMUM RESPECT. 
