I have two different computers but only the one in the picture says it'll be able to run it okay, I dont know much about graphics cards so I really have no idea if the game will play well or not. So can I run the game or do I need to upgrade my graphics card?
The other computer I have is better except for the graphics, it has an Nvidia Geforce 9100.
Well It's an mmo. It's made to run on a wide array of PC's. There is actually an article here on Tom's about that game. I can't find a link but it's there. I think you'll be able to play on the lower side of settings.
In my opinion, no, you will absolutely not be able to run SWTOR on either of those cards. I'm running a 9800m GTS on my notebook, which is better by far than either one of those cards. During the beta weekends I had to put the settings to the absolutely lowest, as well as 1024x768, and the play experience was horrible. It's possible that it was mainly due to overheating and on a desktop you might get better results, but the results will still probably be fairly bad.
If you're thinking of upgrading your card but aren't sure, just buy the game and play it. If it's bad, then you'll know you need to upgrade.
Could someone suggest a good but cheap graphics card? I have just 1 pci slot and a 300 psu.
I hate to say it, but you're probably going to need to upgrade your motherboard so you can run a PCI-E card. And a new power supply. It sounds to me like your computer is just not built to play new-ish 3D games.
That screenshot you provided seems to indicate that you're more than fine, so maybe I'm completely wrong. But I ran that same test on my crappy HP work notebook and it says it would run it too. But I guarantee you it can't. It might launch it, but that's it.