Adobe's MINimums http://www.adobe.com/products/premiere/tech-specs.html
Is that computer 64bit?
What do you plan to edit?
This message has a really good picture showing requirements
Click the picture to expand to readable size - http://forums.adobe.com/thread/810750
More about Requirements http://forums.adobe.com/thread/618058
I plan on edititing things I film, which will be in HD 1080, things like music videos and short videos for school projects nothing longer than a 15 min video. I am not sure about the 64 bit, I will have to ask and get back to you. I am just wondering if this computer will be able to run the Newet version of Adobe without a lot of lag issues?
I would be using my hardrive for the storage of all the media.
[Moved to Hardware forum.]
FYI - All the modern Macs are 64-bit.
Yes, it'll run the software - but I don't think it will do a great job. I believe this is your machine. The processor's so-so (more cores is better), and 8GB of memory is decent (ideally you'd want to upgrade it to 16GB I think), but the Intel HD 3000 graphics is limiting. Intel's integrated graphics tech is pretty good with the 5000 series in the newer 'Haswell' generation of CPU's, before then they were always the least powerful GPU's available.
Shopping around for a used system is a good way to save a buck, but I'd recommend you look for something that has a "discrete" (or dedicated) GPU. Good luck!