[EDIT] There is an issue with this approach, that facebook does not allows sending invite to friends as an app. Instead one needs to do this as a facebook message. So save yourself some time and ignore the facebook part from this tutorial which refers to making requests as an app.
I have been working on a project for some time that is running on Rails 3 and Ruby 1.9.2 and my most recent task was to allow the members to invite their friends via facebook.
I went through some of the tutorials and found the best one that explains it in step by step format. It will surely save me a lot of time and will work as a get reference for me in future as well.
Here is the video, enjoy Image may be NSFW.
Clik here to view.