This is the question many people ask and there's a very simple way to find out. Just follow the steps below and you'll have a good idea whether your computer is able to run the game.
Go to this site: http://www.systemrequirementslab.com/cyri/
using either Internet Explorer or Google Chrome (since it requires ActiveX support) and make sure the browser you're using isn't blocking the plugin.
On the drop down, find "Star Wars: The Old Republic" (it's the last Star Wars title on the list)
Click "Can you run it?" and wait for the results.
If you have any further questions or issues with the game, post up a thread and we'll see if we can help you out.