Okay, I have an extremely outdated ATI Radeon X1270 graphics card (Yes, it sucks, bite me.), and I have a question about updating it. Do I replace the current model folder when I extract it?
Pics provided below since I have no idea how to do this.
A guide for retards like me would be much appreciated.

Just extract wherever it shows by default. After extracting, the installation process should automatically launch.

Double click EXE
Keep clicking next until it stops bugging you
Is this laptop or desktop?

laptop, and it is the correct one (Driver at least).

So what exactly is the problem? Is it refusing to apply the update?

Extract it in a folder. Then run any setups you can find.

From your images, I think that this is a Dell laptop you have. The exe file from Dell is simply a compressed executable. When you execute it (by double clicking on the file), it will show the extraction dialog (the image you have attached). It doesn't matter where you extract it to.
Usually after extraction, the executable is smart enough to automatically start the setup process. If not, navigate to the folder where you extract it to and look for a "setup.exe". Run it and follow any on-screen instructions.