Posting in caps is considered to be shouting and rude, Just so you know
The GX 260 i think came in two different cases so which one is it because if its the thin one that will stand on end or lay flat then you will need a low profile card.
Generally you would put something quite low end with a PC like that. What do you have at the moment ?
First, please turn off caps lock, it will make you sound like a reasonable adult instead of a hyperactive 12 year old.
2nd, do you have a tower (upright) or a desktop/pizzabox (side-lying) GX260?
You have an expansion slot for AGP4x, which is a very old standard, but you can put in an AGP8x card as they are backwards compatable. If you have a desktop/pizzabox style case then be aware that you will need a low profile card, where as the tower version will take a standard card. The fastest card you could possibly put in would be an nVidia 7800 as that was the last generation to use the AGP interface some 8 years ago.
Now, to answer your question (which is relatively irrelivant compared to your particular computer), DDR is a memory standard known as Double Data Rate where it sends data on the tick and the tock of the buss clock (compared to previous technology which could only send one piece of data per clock). GDDR is Graphics DDR, and is a faster wider interface to meet the demand of video cards. This memory is integrated with the card you purchase, and will generally be matched to your GPU so as not to bottleneck it or be overkill for it. So the important thing when picking your graphics card would be to pick it based upon the GPU you need, and the GDDR included will work fine with it.
More important too is how much GDDR ram is included. You will want one with at least 256MB of memory, and if you can find one with 512 then it will be the most possible of that generation.
Lastly, check the power requirements of whatever GPU you purchase as some of them will require more power than your stock power supply.
All of that being said, it may be time to upgrade your platform. Any AGP gpu you find will have a rough time with flash games, much less any video games, or even HD video playback. Yes, it will be better than the integrated graphics, but we are talking a very small difference in the grand scheme of things. $300 can get you a decent 2 year old computer, which will have better (and lower power) graphics than any AGP card you can find. You can reuse your monitor, and may need to upgrade to a USB keyboard and mouse, but a newer tower would be the way to go this time around.